Application of Derivatives

The shortest distance between the point  $\left( {{3 \over 2},0} \right)$   and the curve y = $\sqrt x$, (x > 0), is -

The sum of all the local minimum values of the twice differentiable function f : R $\to$ R defined by $f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)$ is :

The maximum distance from origin of a point on the curve $x = a\sin t - b\sin \left( {{{at} \over b}} \right)$ $y = a\cos t - b\cos \left( {{{at} \over b}} \right),$ both $a,b > 0$ is

The real number $x$ when added to its inverse gives the minimum sum at $x$ equal :

If the function $f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$ where $a>0,$ attains its maximum and minimum at $p$ and $q$ respectively such that ${p^2} = q$ , then $a$ equals

Area of the greatest rectangle that can be inscribed in the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$

The function $f\left( x \right) = {x \over 2} + {2 \over x}$ has a local minimum at

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is

If $p$ and $q$ are positive real numbers such that ${p^2} + {q^2} = 1$, then the maximum value of $(p+q)$ is

Suppose the cubic ${x^3} - px + q$ has three distinct real roots where $p>0$ and $q>0$. Then which one of the following holds?

The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :

Given $P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$ such that $x=0$ is the only real root of $P'\,\left( x \right) = 0.$ If $P\left( { - 1} \right) < P\left( 1 \right),$ then in the interval $\left[ { - 1,1} \right]:$

Let $f:R \to R$ be defined by $f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} & {x \le - 1} \cr {2x + 3,\,\,if} & {x > - 1} \cr } } \right.$ If $f$has a local minimum at $x=-1$, then a possible value of $k$ is

Let $f:R \to R$ be a continuous function defined by $f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}$ Statement - 1 : $f\left( c \right) = {1 \over 3},$ for some $c \in R$. Statement - 2 : $0 < f\left( x \right) \le {1 \over {2\sqrt 2 }},$ for all $x \in R$

For $x \in \left( {0,{{5\pi } \over 2}} \right),$ define $f\left( x \right) = \int\limits_0^x {\sqrt t \sin t\,dt.}$ Then $f$ has

Let $a,b \in R$ be such that the function $f$ given by $f\left( x \right) = In\left| x \right| + b{x^2} + ax,\,x \ne 0$ has extreme values at $x=-1$ and $x=2$ Statement-1 : $f$ has local maximum at $x=-1$ and at $x=2$. Statement-2 : $a = {1 \over 2}$ and $b = {-1 \over 4}$

A line is drawn through the point $(1, 2)$ to meet the coordinate axes at $P$ and $Q$ such that it forms a triangle $OPQ,$ where $O$ is the origin. If the area of the triangle $OPQ$ is least, then the slope of the line $PQ$ is :

If $x=-1$ and $x=2$ are extreme points of $f\left( x \right) = \alpha \,\log \left| x \right|+\beta {x^2} + x$ then

Let $f(x)$ be a polynomial of degree four having extreme values at $x=1$ and $x=2$. If $\mathop {\lim }\limits_{x \to 0} \left[ {1 + {{f\left( x \right)} \over {{x^2}}}} \right] = 3$, then f$(2)$ is equal to :