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JEE Main & Advanced

Lagrange's Mean Value Theorem

  If a function \[f(x)\] is such that,     (i) It is continuous in the closed interval \[[a,b]\]     (ii) It is derivable in the open interval \[(a,\,b)\]     Then there exists at least one value \['c'\] of \[x\] in the open interval \[(a,b)\] such that \[\frac{f(b)-f(a)}{b-a}=f'(c)\].  
Catgeory: JEE Main & Advanced

Rolle's Theorem

 If \[f(x)\]is such that,   (i) It is continuous in the closed interval  \[[a,\,\,b]\]   (ii) It is derivable in the open interval \[(a,\,b)\]   (iii) \[f(a)=f(b)\]   Then there exists at least one value \['c'\] of \[x\] in the open interval \[(a,\,\,b)\] such that \[f'(c)=0\].  
Catgeory: JEE Main & Advanced

Greatest and Least Values of a Function Defined on an Interval \[[a,\,\,b]\]

By maximum (or minimum) or local maximum (or local minimum) value of a function \[f(x)\] at a point \[c\in [a,b]\] we mean the greatest (or the least) value in the immediate neighbourhood of \[x=c\]. It does not mean the greatest or absolute maximum (or the least or absolute minimum) of \[f(x)\]in the interval \[[a,\,b]\].     A function may have a number of local maxima or local minima in a given interval and even a local minimum may be greater than a relative maximum.     Thus a local maximum value may not be the greatest (absolute maximum) and a local minimum value may not be the least (absolute minimum) value of the function in any given interval.     However, if a function \[f(x)\] is continuous on a closed interval \[[a,\,b]\], then it attains the absolute maximum (absolute minimum) at critical points, or at the end points of the more...
Catgeory: JEE Main & Advanced

Properties of Maxima and Minima

(i) Maxima and minima occur alternately, that is between two maxima there is one minimum and vice-versa.     (ii) If \[f(x)\to \infty \]as \[x\to a\] or \[b\] and \[f'(x)=0\] only for one value of \[x\] (say \[c\]) between \[a\] and \[b,\] then \[f(c)\] is necessarily the minimum and the least value.     If \[f(x)\to -\infty \] as \[x\to a\] or \[b,\] then \[f(c)\] is necessarily the maximum and the greatest value.
Catgeory: JEE Main & Advanced

Higher Order Derivative Test

(1) Find \[f'(x)\]and equate it to zero. Solve \[f'(x)=0\]let its roots are \[x={{a}_{1}},{{a}_{2}}\].....     (2) Find  \[{f}''(x)\]and at \[x={{a}_{1}}\];   (i) If \[f''({{a}_{1}})\] is positive, then \[f(x)\] is minimum at \[x={{a}_{1}}\].     (ii) If \[f''({{a}_{1}})\] is negative, then \[f(x)\] is maximum at \[x={{a}_{1}}\].     (iii) If \[f''({{a}_{1}})=0\], go to step 3.     (3) If at \[x={{a}_{1}}\], \[f''({{a}_{1}})=0\], then find \[{f}'''(x)\]. If \[{f}'''({{a}_{1}})\ne 0\], then \[f(x)\]is neither maximum nor minimum at \[x={{a}_{1}}\].     If \[{f}'''({{a}_{1}})=0\], then find \[{{f}^{iv}}(x)\].     If \[{{f}^{iv}}(x)\] is \[+ve\] (Minimum value)     \[{{f}^{iv}}(x)\]is \[-ve\]  (Maximum value)     (4) If at \[x={{a}_{1}},\,\,{{f}^{iv}}({{a}_{1}})=0\], then find \[{{f}^{v}}(x)\] and proceed similarly.  
Catgeory: JEE Main & Advanced

Sufficient Criteria for Extreme Values (1st Derivative Test)

Let \[f(x)\] be a function differentiable at \[x=a\].     Then (a) \[x=a\]is a point of local maximum of \[f(x)\] if     (i) \[f'(a)=0\] and     (ii) \[f'(a)\]changes sign from positive to negative as \[x\] passes through \[a\] i.e., \[f'(x)>0\] at every point in the left neighbourhood \[(a-\delta ,a)\] of \[a\] and \[f'(x)<0\] at every point in the right neighbourhood \[(a,\,\,a+\delta )\] of \[a\].     (b) \[x=a\] is a point of local minimum of \[f(x)\] if     (i) \[f'(a)=0\]and     (ii) \[f'(a)\] changes sign from negative to positive as \[x\] passes through \[a,\] i.e., \[f'(x)<0\] at every point in the left neighbourhood \[(a-\delta ,a)\] of \[a\] and \[{{A}_{1}}=\frac{1}{3}(2a+b),\,{{A}_{2}}=\frac{1}{3}(a+2b)\] at every point in the right neighbourhood \[(a,a+\delta )\]of \[a\].     (c) If \[f'(a)=0\] but \[f'(a)\] does not change sign, that is, has the same sign in the complete neighbourhood of \[a,\] then \[a\] is neither more...
Catgeory: JEE Main & Advanced

Necessary Condition for Extreme Values

 A necessary condition for \[f(a)\]to be an extreme value of a function \[f(x)\]is that \[f'(a)=0\], in case it exists.     Note : (1) This result states that if the derivative exists, it must be zero at the extreme points. A function may however attain an extreme value at a point without being derivable there at.     For example, the function \[f(x)=|x|\] attains the minimum value at the origin even though it is not differentiable at \[x=0\].     (2) This condition is only a necessary condition for the point \[x=a\] to be an extreme point. It is not sufficient i.e., \[f'(a)=0\] does not necessarily imply that \[x=a\] is an extreme point. There are functions for which the derivatives vanish at a point but do not have an extreme value there at e.g. \[f(x)={{x}^{3}}\]at \[x=0\]does not attain an extreme value at \[x=0\] and \[f'(0)=0\].     (3) Geometrically, the more...
Catgeory: JEE Main & Advanced

Definition

(1) A function \[f(x)\] is said to attain a maximum at \[x=a\] if there exists a neighbourhood \[(a-\delta ,a+\delta )\] such that \[f(x)<f(a)\] for all \[x\in (a-\delta ,a+\delta ),x\ne a\]     \[\Rightarrow \]\[f(x)-f(a)<0\] for all \[x\in (a-\delta ,a+\delta ),x\ne a\]     In such a case, \[f(a)\] is said to be the maximum value of \[f'(x)>0\] at \[x=a\].     (2) A function \[f(x)\] is said to attain a minimum at \[x=a\] if there exists a \[nbd\,(a-\delta ,a+\delta )\] such that \[f(x)>f(a)\] for all \[x\in (a-\delta ,a+\delta ),x\ne a\]     \[\Rightarrow \] \[f(x)-f(a)>0\] for all \[x\in (a-\delta ,a+\delta ),x\ne a\]     In such a case, \[f(a)\]is said to be the minimum value of \[f(x)\] at \[x=a\]. The points at which a function attains either the maximum values or the minimum values are known as the extreme points or turning points and both maximum and minimum values of more...
Catgeory: JEE Main & Advanced

Definition

    (1) A function \[f\] is said to be an increasing function in \[]\,a,\,b[,\] if \[{{x}_{1}}<{{x}_{2}}\Rightarrow f({{x}_{1}})<f({{x}_{2}})\] for all \[{{x}_{1}},{{x}_{2}}\in \,]\,a,b\,[.\]     (2) A function \[f\] is said to be a decreasing function in \[]\,a,\,b[,\] if \[{{x}_{1}}<{{x}_{2}}\Rightarrow f({{x}_{1}})>f({{x}_{2}})\], \[\,\,{{x}_{1}},{{x}_{2}}\in \,]\,a,\,b\,[.\]    · \[f(x)\]is known as non-decreasing if \[f'(x)\ge 0\] and non-increasing if \[f'(x)\le 0\].     Monotonic function :  A function \[f\] is said to be monotonic in an interval if it is either increasing or decreasing in that interval.     We summarize the results in the table below :                  
\[{f}'(\,{{a}_{1}})\] \[{f}''(\,{{a}_{1}})\] more...
Catgeory: JEE Main & Advanced

Length of Perpendicular from Origin to the Tangent

  Length of perpendicular from origin \[(0,\,0)\] to the tangent drawn at point \[P({{x}_{1}},\,{{y}_{1}})\] of the curve \[y=f(x)\] \[p=\left| \,\frac{{{y}_{1}}-{{x}_{1}}{{\left( \frac{dy}{dx} \right)}_{({{x}_{1}},\,{{y}_{1}})}}}{\sqrt{1+{{\left( \frac{dy}{dx} \right)}^{2}}}}\, \right|\] .  
Catgeory: JEE Main & Advanced

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