JEE Main & Advanced Mathematics Applications of Derivatives Question Bank

done Maxima and Minima Total Questions - 99

• question_answer1)
  The value of the function \[(x-1){{(x-2)}^{2}}\] at its maxima is
  A)
             1 done clear
  B)
             2 done clear
  C)
             0 done clear
  D)
             \[\frac{4}{27}\] done clear
  View Solution play_arrow

• question_answer2)
  The maximum and minimum values of the function \[|\sin 4x+3|\]are
  A)
             1, 2 done clear
  B)
             4, 2 done clear
  C)
             2, 4 done clear
  D)
             ? 1, 1 done clear
  View Solution play_arrow

• question_answer3)
  Local maximum and local minimum values of the function \[(x-1){{(x+2)}^{2}}\]are
  A)
             ? 4, 0 done clear
  B)
             0, ? 4 done clear
  C)
             4, 0 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer4)
  The function \[{{x}^{5}}-5{{x}^{4}}+5{{x}^{3}}-10\]has a maximum, when x = [MP PET 1995]
  A)
             3 done clear
  B)
             2 done clear
  C)
             1 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer5)
  The maximum value of function \[{{x}^{3}}-12{{x}^{2}}+36x+\]17  in the interval [1, 10] is
  A)
             17 done clear
  B)
             177 done clear
  C)
             77 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer6)
  The maximum value of the function \[{{x}^{3}}+{{x}^{2}}+x-4\] is
  A)
             127 done clear
  B)
             4 done clear
  C)
             Does not have a maximum value done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer7)
  The function \[{{x}^{2}}\log x\]in the interval (1, e) has
  A)
             A point of maximum          done clear
  B)
             A point of minimum           done clear
  C)
             Points of maximum as well as of minimum done clear
  D)
             Neither a point of maximum nor minimum done clear
  View Solution play_arrow

• question_answer8)
  The minimum value of \[|x|+|x+\frac{1}{2}|+|x-3|+|x-\frac{5}{2}|\] is
  A)
             0 done clear
  B)
             2 done clear
  C)
             4 done clear
  D)
             6 done clear
  View Solution play_arrow

• question_answer9)
  Local maximum value of the function \[\frac{\log x}{x}\]is [MNR 1984; RPET 1997, 2002; DCE 2002; Karnataka CET 2000; UPSEAT 2001; MP PET 2002]
  A)
             e done clear
  B)
             1 done clear
  C)
                  \[\frac{1}{e}\] done clear
  D)
                  2e done clear
  View Solution play_arrow

• question_answer10)
  If \[x+y=16\] and \[{{x}^{2}}+{{y}^{2}}\] is minimum, then the values of x and y are
  A)
             3, 13 done clear
  B)
             4, 12 done clear
  C)
             6, 10 done clear
  D)
             8, 8 done clear
  View Solution play_arrow

• question_answer11)
  The function \[x\sqrt{1-{{x}^{2}}},(x>0)\]has
  A)
             A local maxima                     done clear
  B)
             A local minima done clear
  C)
             Neither a local maxima nor a local minima done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer12)
  If two sides of a triangle be given, then the area of the triangle will be maximum if the angle between  the given sides be
  A)
             \[\frac{\pi }{3}\] done clear
  B)
             \[\frac{\pi }{4}\] done clear
  C)
             \[\frac{\pi }{6}\] done clear
  D)
             \[\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer13)
  The function \[{{x}^{5}}-5{{x}^{4}}+5{{x}^{3}}-1\]is [MP PET 1993]
  A)
             Maximum at \[x=3\]and minimum at \[x=1\] done clear
  B)
             Minimum at \[x=1\] done clear
  C)
             Neither maximum nor minimum at \[x=0\] done clear
  D)
             Maximum at \[x=0\] done clear
  View Solution play_arrow

• question_answer14)
  The adjacent sides of a rectangle with given perimeter as 100 cm and enclosing maximum area are  [MP PET 1993]
  A)
             10 cm and 40 cm done clear
  B)
             20 cm and 30 cm done clear
  C)
              25 cm and 25 cm done clear
  D)
    15 cm and 35 cm done clear
  View Solution play_arrow

• question_answer15)
  The necessary condition to be maximum or minimum for the function is
  A)
             \[f'(x)=0\]and it is sufficient done clear
  B)
             \[f''(x)=0\]and it is sufficient done clear
  C)
             \[f'(x)=0\]but it is not sufficient done clear
  D)
             \[f'(x)=0\]and \[f''(x)=-ve\] done clear
  View Solution play_arrow

• question_answer16)
  The area of a rectangle will be maximum for the given perimeter, when rectangle is a [AI CBSE 1991; RPET 1999]
  A)
             Parallelogram done clear
  B)
             Trapezium done clear
  C)
             Square done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer17)
  Of the given perimeter, the triangle having maximum area is
  A)
             Isosceles triangle done clear
  B)
             Right angled triangle done clear
  C)
             Equilateral done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer18)
  The sufficient conditions for the function \[f:R\to R\] is to be maximum at \[x=a\], will be
  A)
             \[f'(a)>0\]and \[f''(a)>0\]        done clear
  B)
             \[f'(a)=0\]and \[f''(a)=0\] done clear
  C)
             \[f'(a)=0\]and \[f''(a)<0\]        done clear
  D)
             \[f'(a)>0\]and \[f''(a)<0\] done clear
  View Solution play_arrow

• question_answer19)
  36 factorize into two factors in such a way that sum of factors is minimum, then the factors are                  [MP PET 1987]
  A)
             2, 18 done clear
  B)
             9, 4 done clear
  C)
             3, 12 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer20)
  If \[f(x)=2{{x}^{3}}-3{{x}^{2}}-12x+5\]and \[x\in [-2,\,4]\], then the maximum value of function is at the following value of x                          [MP PET 1987, 2000; Orissa JEE 2005]
  A)
             2 done clear
  B)
             ?1 done clear
  C)
             ? 2 done clear
  D)
             4 done clear
  View Solution play_arrow

• question_answer21)
  The point for the curve \[y=x{{e}^{x}}\]                                                [MNR 1990]
  A)
             \[x=-1\]is minimum done clear
  B)
             \[x=0\]is minimum done clear
  C)
             \[x=-1\]is maximum done clear
  D)
              \[x=0\]is maximum done clear
  View Solution play_arrow

• question_answer22)
  The function \[\sin x(1+\cos x)\]at \[x=\frac{\pi }{3}\], is
  A)
             Maximum                              done clear
  B)
             Minimum done clear
  C)
             Neither maximum nor minimum done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer23)
  Maximum value of \[{{\left( \frac{1}{x} \right)}^{x}}\] is  [DCE 1999; Karnataka CET 1999; UPSEAT 2003]
  A)
             \[{{(e)}^{e}}\]                      done clear
  B)
             \[{{(e)}^{e}}\] done clear
  C)
             \[{{(e)}^{-e}}\] done clear
  D)
             \[{{\left( \frac{1}{e} \right)}^{e}}\] done clear
  View Solution play_arrow

• question_answer24)
  If \[x+y=10\], then the maximum value of xy is
  A)
             5 done clear
  B)
             20 done clear
  C)
             25 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer25)
  The sum of two numbers is fixed. Then its multiplication is maximum, when
  A)
             Each number is half of the sum done clear
  B)
             Each number is \[\frac{1}{3}\]and  \[\frac{2}{3}\] respectively of the sum done clear
  C)
             Each number is \[\frac{1}{4}\] and \[\frac{3}{4}\] respectively of the sum done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer26)
  The two parts of 100 for which the sum of double of first and square of second part is minimum, are
  A)
             50, 50 done clear
  B)
             99, 1 done clear
  C)
             98, 2 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer27)
  The number that exceeds its square by the greatest amount is                                                  [Roorkee 1990]
  A)
             ? 1 done clear
  B)
             0 done clear
  C)
             \[\frac{1}{2}\] done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer28)
  If for a function \[f(x),f'(a)=0,f''(a)=0\], \[{{f}''}'(a)>0\],  then at \[x=a\], \[f(x)\] is     [MP PET 1994; Pb. CET 2002]
  A)
             Minimum done clear
  B)
             Maximum done clear
  C)
             Not an extreme point done clear
  D)
             Extreme point done clear
  View Solution play_arrow

• question_answer29)
  The least value of the sum of any positive real number and its reciprocal is                                            [MP PET 1994]
  A)
             1 done clear
  B)
             2 done clear
  C)
             3 done clear
  D)
             4 done clear
  View Solution play_arrow

• question_answer30)
  \[{{x}^{x}}\] has a stationary point at                                                                      [Karnataka CET 1993]
  A)
             \[x=e\] done clear
  B)
             \[x=\frac{1}{e}\] done clear
  C)
             \[x=1\] done clear
  D)
             \[x=\sqrt{e}\] done clear
  View Solution play_arrow

• question_answer31)
  OR When x is positive, the minimum value of \[{{x}^{x}}\]is [EAMCET 1987]
  A)
             \[{{e}^{-1}}\]                        done clear
  B)
             \[{{e}^{-1/e}}\] done clear
  C)
             \[{{e}^{1/e}}\] done clear
  D)
             \[{{e}^{e}}\] done clear
  View Solution play_arrow

• question_answer32)
  The value of a so that the sum of the squares of the roots of the equation \[{{x}^{2}}-(a-2)x-a+1=0\] assume the least value, is  [RPET 2000; AIEEE 2005]
  A)
             2 done clear
  B)
             1 done clear
  C)
             3 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer33)
  The largest term in the sequence \[{{a}_{n}}=\frac{{{n}^{2}}}{{{n}^{3}}+200}\] is given by
  A)
             \[\frac{529}{49}\] done clear
  B)
             \[\frac{8}{89}\] done clear
  C)
             \[\frac{49}{543}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer34)
  The maximum value of xy subject to \[x+y=8\], is                    [MNR 1995]
  A)
             8 done clear
  B)
             16 done clear
  C)
             20 done clear
  D)
             24 done clear
  View Solution play_arrow

• question_answer35)
  A minimum value of \[\int_{0}^{x}{t{{e}^{-{{t}^{2}}}}}\]dt  is       [EAMCET 2003]
  A)
             1 done clear
  B)
             2 done clear
  C)
             3 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer36)
  If sum of two numbers is 3, then maximum value of the product of first and the square of second is [MP PET 1996]
  A)
             4 done clear
  B)
             3 done clear
  C)
             2 done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer37)
  The minimum value of the function \[2\cos 2x-\cos 4x\]in \[0\le x\le \pi \]is
  A)
             0 done clear
  B)
             1 done clear
  C)
             \[\frac{3}{2}\] done clear
  D)
             ? 3 done clear
  View Solution play_arrow

• question_answer38)
  If \[f(x)=2{{x}^{3}}-21{{x}^{2}}+36x-30\], then which one of the following is correct
  A)
             \[f(x)\] has minimum at \[x=1\] done clear
  B)
             \[f(x)\] has maximum at \[x=6\] done clear
  C)
             \[f(x)\]has maximum at \[x=1\] done clear
  D)
             \[f(x)\] has no maxima or minima done clear
  View Solution play_arrow

• question_answer39)
  The maximum value of \[2{{x}^{3}}-24x+107\] in the interval       [?3, 3] is
  A)
             75 done clear
  B)
             89 done clear
  C)
             125 done clear
  D)
             139 done clear
  View Solution play_arrow

• question_answer40)
  If the function \[f(x)={{x}^{4}}-62{{x}^{2}}+ax+9\]is maximum at \[x=1\], then the value of a is
  A)
             120 done clear
  B)
             ?120 done clear
  C)
             52 done clear
  D)
             128 done clear
  View Solution play_arrow

• question_answer41)
  The minimum value of the expression \[7-20x+11{{x}^{2}}\] is
  A)
             \[\frac{177}{11}\] done clear
  B)
    \[-\frac{177}{11}\] done clear
  C)
   \[-\frac{23}{11}\] done clear
  D)
             \[\frac{23}{11}\] done clear
  View Solution play_arrow

• question_answer42)
  Maximum value of \[x{{(1-x)}^{2}}\] when \[0\le x\le 2\], is [MP PET 1997]
  A)
             \[\frac{2}{27}\]                    done clear
  B)
             \[\frac{4}{27}\] done clear
  C)
             5     done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer43)
  If from a wire of length 36 metre a rectangle of greatest area is made, then its two adjacent sides in metre are                                                                                            [MP PET 1998]
  A)
             6, 12 done clear
  B)
             9, 9 done clear
  C)
             10, 8 done clear
  D)
             13, 5 done clear
  View Solution play_arrow

• question_answer44)
  The minimum value of \[2{{x}^{2}}+x-1\]is    [EAMCET 2003]
  A)
   \[-\frac{1}{4}\] done clear
  B)
             \[\frac{3}{2}\] done clear
  C)
    \[\frac{-9}{8}\] done clear
  D)
             \[\frac{9}{4}\] done clear
  View Solution play_arrow

• question_answer45)
  The minimum value of the function \[y=2{{x}^{3}}-21{{x}^{2}}+36x-20\] is                            [MP PET 1999]
  A)
             ?128 done clear
  B)
             ?126 done clear
  C)
             ?120 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer46)
  The sum of two non-zero numbers is 4. The minimum value of the sum of their reciprocals is        [Kurukshetra CEE 1998]
  A)
             \[\frac{3}{4}\] done clear
  B)
             \[\frac{6}{5}\] done clear
  C)
             1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer47)
  The minimum value of \[[(5+x)(2+x)]/[1+x]\] for non-negative real x is                  [Kurukshetra  CEE 1998]
  A)
             12 done clear
  B)
             1 done clear
  C)
             9 done clear
  D)
             8 done clear
  View Solution play_arrow

• question_answer48)
  One maximum point of \[{{\sin }^{p}}x{{\cos }^{q}}x\]is [RPET 1997; AMU 2000]
  A)
             \[x={{\tan }^{-1}}\sqrt{(p/q)}\] done clear
  B)
             \[x={{\tan }^{-1}}\sqrt{(q/p)}\] done clear
  C)
             \[x={{\tan }^{-1}}(p/q)\] done clear
  D)
             \[x={{\tan }^{-1}}(q/p)\] done clear
  View Solution play_arrow

• question_answer49)
  20 is divided into two parts so that product of cube of one quantity and square of the other quantity is maximum. The parts are [RPET 1997]
  A)
             10, 10 done clear
  B)
             16, 4 done clear
  C)
             8, 12 done clear
  D)
             12, 8 done clear
  View Solution play_arrow

• question_answer50)
  If \[f(x)=\frac{{{x}^{2}}-1}{{{x}^{2}}+1}\], for every real number x, then the minimum value of f                [IIT 1998]
  A)
             Does not exist because f is unbounded done clear
  B)
             Is not attained even though f is bounded done clear
  C)
             Is equal to 1 done clear
  D)
             Is equal to ?1 done clear
  View Solution play_arrow

• question_answer51)
  The number of values of x where the function \[f(x)=\cos x+\cos (\sqrt{2}x)\] attains its maximum is [IIT 1998; DCE 2001, 05]
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             Infinite done clear
  View Solution play_arrow

• question_answer52)
  The minimum value of \[{{e}^{(2{{x}^{2}}-2x+1){{\sin }^{2}}x}}\] is [Roorkee Qualifying 1998]
  A)
             e done clear
  B)
             1/e done clear
  C)
             1 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer53)
  x  and y be two variables such that \[x>0\] and\[xy=1\]. Then the minimum value of \[x+y\] is [Kurukshetra CEE 1998; MP PET 2002]
  A)
             2 done clear
  B)
             3 done clear
  C)
             4 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer54)
  What are the minimum and maximum values of the function \[{{x}^{5}}-5{{x}^{4}}+5{{x}^{3}}-10\]                                     [DCE 1999]
  A)
             ? 37, ? 9 done clear
  B)
             10, 0 done clear
  C)
             It has 2 min. and 1 max. values done clear
  D)
             It has 2 max. and 1 min. values done clear
  View Solution play_arrow

• question_answer55)
  Divide 20 into two parts such that the product of one part and the cube of the other is maximum. The two parts are                                                                       [DCE 1999]
  A)
             (10, 10) done clear
  B)
             \[(5,\,\,15)\] done clear
  C)
             (13, 7) done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer56)
  The maximum and minimum values of \[{{x}^{3}}-18{{x}^{2}}+96x\] in interval (0, 9) are                                                [RPET 1999]
  A)
             160, 0 done clear
  B)
             60, 0 done clear
  C)
             160, 128 done clear
  D)
             120, 28 done clear
  View Solution play_arrow

• question_answer57)
  The maximum value of \[\sin x\,\,(1+\cos x)\] will be at the                                                                                            [UPSEAT 1999]
  A)
             \[x=\frac{\pi }{2}\] done clear
  B)
             \[x=\frac{\pi }{6}\] done clear
  C)
             \[x=\frac{\pi }{3}\] done clear
  D)
             \[x=\pi \] done clear
  View Solution play_arrow

• question_answer58)
  \[\frac{x}{1+x\,\tan x}\] is maxima at                                                                     [UPSEAT 1999]
  A)
             \[x=\sin x\] done clear
  B)
             \[x=\cos x\] done clear
  C)
             \[x=\frac{\pi }{3}\] done clear
  D)
             \[x=\tan x\] done clear
  View Solution play_arrow

• question_answer59)
  If x is real, then greatest and least values of \[\frac{{{x}^{2}}-x+1}{{{x}^{2}}+x+1}\] are            [RPET 1999; AMU 1999; UPSEAT 2002]
  A)
             \[3,\,-\frac{1}{2}\] done clear
  B)
             \[3,\frac{1}{3}\] done clear
  C)
             \[-3,\,-\frac{1}{3}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer60)
   The minimum value of \[\frac{\log x}{x}\] in the interval \[[2,\,\infty )\] is                                                                       [Roorkee 1999]
  A)
             \[\frac{\log 2}{2}\] done clear
  B)
             Zero done clear
  C)
             \[\frac{1}{e}\] done clear
  D)
             Does not exist done clear
  View Solution play_arrow

• question_answer61)
  The maximum value of \[{{x}^{4}}{{e}^{-{{x}^{2}}}}\]is                   [AMU 1999]
  A)
             \[{{e}^{2}}\] done clear
  B)
             \[{{e}^{-2}}\] done clear
  C)
             \[12{{e}^{-2}}\] done clear
  D)
             \[4{{e}^{-2}}\] done clear
  View Solution play_arrow

• question_answer62)
  If \[A+B=\frac{\pi }{2},\] the maximum value of \[\cos A\cos B\]is [AMU 1999]
  A)
             \[\frac{1}{2}\] done clear
  B)
             \[\frac{3}{4}\] done clear
  C)
             1 done clear
  D)
             \[\frac{4}{3}\] done clear
  View Solution play_arrow

• question_answer63)
  The real number x when added to its inverse gives the minimum value of the sum at x equal to [RPET 2000; AIEEE 2003]
  A)
             ? 2 done clear
  B)
             2 done clear
  C)
             1 done clear
  D)
             ? 1 done clear
  View Solution play_arrow

• question_answer64)
  The denominator of a fraction number is greater than 16 of the square of numerator, then least value of the number is                                                                      [RPET 2000]
  A)
             \[-1/4\] done clear
  B)
             \[-1/8\] done clear
  C)
             \[1/12\] done clear
  D)
             \[1/16\] done clear
  View Solution play_arrow

• question_answer65)
  The real number which most exceeds its cube is [MP PET 2000]
  A)
             \[\frac{1}{2}\] done clear
  B)
             \[\frac{1}{\sqrt{3}}\] done clear
  C)
             \[\frac{1}{\sqrt{2}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer66)
  The maximum value of \[f(x)=\frac{x}{4+x+{{x}^{2}}}\] on \[[-1,\,1]\] is                                                                       [MP PET 2000]
  A)
             \[-1/4\] done clear
  B)
             \[-1/3\] done clear
  C)
             \[1/6\] done clear
  D)
             \[1/5\] done clear
  View Solution play_arrow

• question_answer67)
  A cone of maximum volume is inscribed in a given sphere, then ratio of the height of the cone to diameter of the sphere is [MNR 1985; UPSEAT 2000]
  A)
             2/3 done clear
  B)
             3/4 done clear
  C)
             1/3 done clear
  D)
             ¼ done clear
  View Solution play_arrow

• question_answer68)
   The ratio of height of cone of maximum volume inscribed in a sphere to its radius is                        [Orissa JEE  2004]
  A)
             \[\frac{3}{4}\] done clear
  B)
             \[\frac{4}{3}\] done clear
  C)
             \[\frac{1}{2}\] done clear
  D)
             \[\frac{2}{3}\] done clear
  View Solution play_arrow

• question_answer69)
  The function \[f(x)=x+\sin x\] has  [AMU 2000]
  A)
             A minimum but no maximum done clear
  B)
             A maximum but no minimum done clear
  C)
             Neither maximum nor minimum done clear
  D)
             Both maximum and minimum done clear
  View Solution play_arrow

• question_answer70)
  The function \[f(x)=ax+\frac{b}{x};a,\,b,x>0\] takes on the least value at x equal to         [AMU 2000]
  A)
             b done clear
  B)
             \[\sqrt{a}\] done clear
  C)
              \[\sqrt{b}\] done clear
  D)
              \[\sqrt{b/a}\] done clear
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• question_answer71)
  If \[xy={{c}^{2}},\] then minimum value of \[ax+by\] is [RPET 2001]
  A)
             \[c\sqrt{ab}\] done clear
  B)
             \[2c\sqrt{ab}\] done clear
  C)
             \[-c\sqrt{ab}\] done clear
  D)
             \[-2c\sqrt{ab}\] done clear
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• question_answer72)
  If \[{{a}^{2}}{{x}^{4}}+{{b}^{2}}{{y}^{4}}={{c}^{6}},\] then maximum value of xy is                                                                                                [RPET 2001]
  A)
             \[\frac{{{c}^{2}}}{\sqrt{ab}}\] done clear
  B)
             \[\frac{{{c}^{3}}}{ab}\] done clear
  C)
              \[\frac{{{c}^{3}}}{\sqrt{2ab}}\] done clear
  D)
              \[\frac{{{c}^{3}}}{2ab}\] done clear
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• question_answer73)
  The function \[f(x)=2{{x}^{3}}-15{{x}^{2}}+36x+4\] is maximum at                                                                          [Karnataka CET 2001]
  A)
             \[x=2\] done clear
  B)
             \[x=4\] done clear
  C)
             \[x=0\] done clear
  D)
             \[x=3\] done clear
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• question_answer74)
  Maximum slope of the curve \[y=-{{x}^{3}}+3{{x}^{2}}+9x-27\] is                                                                           [MP PET 2001]
  A)
             0 done clear
  B)
             12 done clear
  C)
             16 done clear
  D)
             32 done clear
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• question_answer75)
  The function \[f(x)=2{{x}^{3}}-3{{x}^{2}}-12x+4\] has [DCE 2002]
  A)
             No maxima and minima done clear
  B)
             One maximum and one minimum done clear
  C)
             Two maxima done clear
  D)
             Two minima done clear
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• question_answer76)
  If \[f(x)=\frac{1}{4{{x}^{2}}+2x+1}\], then its maximum value is                                                                                                [RPET 2002]
  A)
             4/3 done clear
  B)
             2/3 done clear
  C)
             1     done clear
  D)
             ¾ done clear
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• question_answer77)
  If \[f(x)=x+\frac{1}{x},\] x > 0, then its greatest value is [RPET 2002]
  A)
             ? 2 done clear
  B)
             0 done clear
  C)
             3 done clear
  D)
             None of these done clear
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• question_answer78)
  The perimeter of a sector is p. The area of the sector is maximum when its radius is                                         [Karnataka CET 2002]
  A)
             \[\sqrt{p}\] done clear
  B)
             \[\frac{1}{\sqrt{p}}\] done clear
  C)
             \[\frac{p}{2}\] done clear
  D)
             \[\frac{p}{4}\] done clear
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• question_answer79)
  v If \[y=a\,\,\log x+b{{x}^{2}}+x\]has its extremum value at \[x=1\]and \[x=2,\]then \[(a,b)\]=                                     [UPSEAT 2002]
  A)
             \[\left( 1,\,\,\frac{1}{2} \right)\] done clear
  B)
             \[\left( \frac{1}{2},\,2 \right)\] done clear
  C)
              \[\left( 2,\,\,\frac{-1}{2} \right)\] done clear
  D)
              \[\left( \frac{-2}{3},\,\,\frac{-1}{6} \right)\] done clear
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• question_answer80)
  In \[(-4,\,4)\] the function \[f(x)=\int\limits_{-10}^{x}{({{t}^{4}}-4){{e}^{-4t}}dt}\] has [AMU 2002]
  A)
              No extrema done clear
  B)
              One extremum done clear
  C)
              Two extrema done clear
  D)
              Four extrema done clear
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• question_answer81)
  On [1, e] the greatest value of \[{{x}^{2}}\log x\]   [AMU 2002]
  A)
             \[{{e}^{2}}\] done clear
  B)
             \[\frac{1}{e}\log \frac{1}{\sqrt{e}}\] done clear
  C)
             \[{{e}^{2}}\log \sqrt{e}\] done clear
  D)
             None of these done clear
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• question_answer82)
  The function \[f(x)={{x}^{-x}},\,(x\,\in \,R)\] attains a maximum value at x =       [EAMCET 2002]
  A)
             2 done clear
  B)
             3 done clear
  C)
             1/e done clear
  D)
             1 done clear
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• question_answer83)
  If \[ab=2a+3b,\,a>0,\,\,b>0\] then the minimum value of ab is                                                  [Orissa JEE 2002]
  A)
             12 done clear
  B)
             24 done clear
  C)
             \[\frac{1}{4}\] done clear
  D)
             None of these done clear
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• question_answer84)
  If PQ  and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle is                                                                [Kerala (Engg.) 2002]
  A)
             \[\pi \] done clear
  B)
             \[\pi /3\] done clear
  C)
             \[\pi /4\] done clear
  D)
             \[\pi /2\] done clear
  View Solution play_arrow

• question_answer85)
  The function \[y=a(1-\cos x)\] is maximum when \[x=\]                                                                                   [Kerala (Engg.) 2002]
  A)
             \[\pi \] done clear
  B)
             \[\pi /2\] done clear
  C)
             \[-\pi /2\] done clear
  D)
             \[-\pi /6\] done clear
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• question_answer86)
  The minimum value of \[\left( {{x}^{2}}+\frac{250}{x} \right)\] is [Kurukshetra CEE 2002]
  A)
             75 done clear
  B)
             50 done clear
  C)
             25 done clear
  D)
             55 done clear
  View Solution play_arrow

• question_answer87)
  The minimum value of \[{{x}^{2}}+\frac{1}{1+{{x}^{2}}}\] is at    [UPSEAT 2003]
  A)
             \[x=0\] done clear
  B)
             \[x=1\] done clear
  C)
             \[x=4\] done clear
  D)
             \[x=3\] done clear
  View Solution play_arrow

• question_answer88)
  If \[x-2y=4,\] the minimum value of \[xy\] is [UPSEAT 2003]
  A)
             ? 2 done clear
  B)
             2 done clear
  C)
             0 done clear
  D)
             ? 3 done clear
  View Solution play_arrow

• question_answer89)
  The minimum value of \[2x+3y,\] when \[xy=6,\] is [MP PET 2003]
  A)
             12 done clear
  B)
             9 done clear
  C)
             8 done clear
  D)
             6 done clear
  View Solution play_arrow

• question_answer90)
  The function \[f(x)=|px-q|+r|x|,\] \[x\in (-\infty ,\infty )\] where \[p>0,q>0,r>0\]assumes its minimum value only at one point if [Pb. CET 2003]
  A)
             \[p\ne q\] done clear
  B)
             \[q\ne r\] done clear
  C)
             \[r\ne p\] done clear
  D)
             \[p=q=r\] done clear
  View Solution play_arrow

• question_answer91)
  The minimum value of function \[f(x)=3{{x}^{4}}-8{{x}^{3}}+12{{x}^{2}}-48x+25\] on [0, 3] is equal to  [Pb. CET 2004]
  A)
             25 done clear
  B)
             ? 39 done clear
  C)
             ? 25 done clear
  D)
             39 done clear
  View Solution play_arrow

• question_answer92)
  The maximum value of \[{{x}^{1/x}}\] is                                                [MP PET 2004]
  A)
             \[\frac{1}{e}\] done clear
  B)
             \[{{e}^{1/e}}\] done clear
  C)
             e done clear
  D)
             \[\frac{1}{{{e}^{e}}}\] done clear
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• question_answer93)
  The minimum value of \[4{{e}^{2x}}+9{{e}^{-2x}}\] is  [J & K 2005]
  A)
             11 done clear
  B)
             12 done clear
  C)
             10 done clear
  D)
             14 done clear
  View Solution play_arrow

• question_answer94)
  The point \[(0,\,5)\]is closest to the curve \[{{x}^{2}}=2y\] at            [MNR 1983]
  A)
             \[(2\sqrt{2},0)\] done clear
  B)
             (0, 0) done clear
  C)
             \[(2,\,2)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer95)
  The maximum value of xy when \[x+2y=8\] is            [Kerala (Engg.) 2005]
  A)
             20 done clear
  B)
             16 done clear
  C)
             24 done clear
  D)
             8 done clear
  E)
             4 done clear
  View Solution play_arrow

• question_answer96)
  The minimum value of \[P(1,\,1)\] is                                                               [DCE 2005]
  A)
             \[\frac{15}{2}\]                    done clear
  B)
             \[\frac{11}{2}\] done clear
  C)
             \[\frac{-13}{2}\] done clear
  D)
             \[\frac{71}{8}\] done clear
  View Solution play_arrow

• question_answer97)
  If \[P=(1,\,1)\], \[Q=(3,\,2)\] and R is a point on x-axis  then the value of \[PR+RQ\] will be minimum at                 [AMU 2005]
  A)
             \[\left( \frac{5}{3},\,0 \right)\] done clear
  B)
             \[\left( \frac{1}{3},\,0 \right)\] done clear
  C)
             (3, 0) done clear
  D)
             (1, 0) done clear
  View Solution play_arrow

• question_answer98)
  Let \[f\,(x)=1+2{{x}^{2}}+{{2}^{2}}{{x}^{4}}+.....+{{2}^{10}}{{x}^{20}}\],  then \[f(x)\] has                                                               [AMU 2005]
  A)
             More than one minimum done clear
  B)
             Exactly one minimum done clear
  C)
             At least one maximum done clear
  D)
             None of these done clear
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• question_answer99)
  Area of the greatest rectangle that can be inscribed in the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is                                       [AIEEE 2005]                                         
  A)
             \[\sqrt{ab}\] done clear
  B)
             \[\frac{a}{b}\] done clear
  C)
                  \[2ab\] done clear
  D)
                  \[ab\] done clear
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