# Current Affairs 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 $\frac{f(b)-f(a)}{b-a}=f'(c)$.

#### 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 $f'(c)=0$.

#### 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 interval $[a,\,b]$. Thus, to find the absolute maximum (absolute minimum) value of the function, we choose the largest and smallest amongst the numbers $f(a),f({{c}_{1}}),f({{c}_{2}}),....,f({{c}_{n}}),f(b)$, where $x={{c}_{1}},{{c}_{2}},....,{{c}_{n}}$ are the critical points.

#### 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.

#### 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.

#### 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 a point of local maximum nor a point of local minimum.     Working rule for determining extreme values of a function $f(x)$     Step I : Put $y=f(x)$     Step II : Find $\frac{dy}{dx}$     Step III : Put $\frac{dy}{dx}=0$ and solve this equation for $x$. Let $x={{c}_{1}},{{c}_{2}},.....,{{c}_{n}}$ be values of $x$ obtained by putting $\frac{dy}{dx}=0.$ ${{c}_{1}},\,{{c}_{2}},\,.........{{c}_{n}}$ are the stationary values of $x$.     Step IV : Consider $x={{c}_{1}}$.     If $\frac{dy}{dx}$ changes its sign from positive to negative as $x$ passes through ${{c}_{1}}$, then the function attains a local maximum at $x={{c}_{1}}$. If $\frac{dy}{dx}$ changes its sign from negative to positive as $x$ passes through ${{c}_{1}}$, then the function attains a local minimum at $x={{c}_{1}}$. In case there is no change of sign, then $x={{c}_{1}}$ is neither a point of local maximum nor a point of local minimum.

#### 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 above condition means that the tangent to the curve $y=f(x)$ at a point where the ordinate is maximum or minimum is parallel to the x-axis.     (4) The values of $x$ for which $f'(x)=0$ are called stationary values or critical values of $x$ and the corresponding values of $f(x)$ are called stationary or turning values of $f(x)$.     (5) The points where a function attains a maximum (or minimum)  are also known as points of local maximum (or local minimum) and the corresponding values of $f(x)$ are called local maximum (or local minimum) values.

#### 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 $f(x)$ are called extreme or extreme values.     Thus a function attains an extreme value at $x=a$ if $f(a)$is either a maximum or a minimum value. Consequently at an extreme point $a,\,\,f(x)-f(a)$ keeps the same sign for all values of $x$ in a deleted $nbd$of $a$.

#### 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 :                   more...

#### 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 $y=f(x)$ $p=\left| \,\frac{{{y}_{1}}-{{x}_{1}}{{\left( \frac{dy}{dx} \right)}_{({{x}_{1}},\,{{y}_{1}})}}}{\sqrt{1+{{\left( \frac{dy}{dx} \right)}^{2}}}}\, \right|$ .

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 ${f}'(\,{{a}_{1}})$ ${f}''(\,{{a}_{1}})$ ${f}'''(\,{{a}_{1}})$ Behaviour of $f$ at ${{a}_{1}}$ + Increasing $-$ Decreasing 0 + Minimum 0 $-$ Maximum