Let $f\left(x\right)=1+2x^2+2^2{x}^4+.\dots \dots +2^{10}{x}^{20}$.Then, $f\left(x\right)$ has

Important Questions on Application of Derivatives

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

From the top of a $64$ metres high tower, a stone is thrown upwards vertically with the velocity of $48 \mathrm{m}/\mathrm{s}.$ The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration $g=32 \mathrm{m}/{\mathrm{s}}^2$, is:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If a right circular cone, having maximum volume, is inscribed in a sphere of radius $3 cm$, then the curved surface area (in ${\mathrm{cm}}^2$) of this cone is :

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum volume $\left(\text{in}\mathrm{ }\mathrm{cubic} \mathrm{m}\right)$ of the right circular cone having slant height $3 \mathrm{m}$ is:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum value of $f\left(x\right)=\frac{x}{4+x+x^2}$ on $[-1, 1]$ is

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f\left(x\right)=9x^4+12x^3-36x^2+25,x\in R,$ then

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum area (in sq. units) of a rectangle having its base on the $x-$ axis and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola, is :

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If $f\left(x\right)=\frac{x}{x^2+1}$ is an increasing function then the value of $x$ lies in

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum area of a rectangle that can be inscribed in a circle of radius $2$ units is

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If $f\left(x\right)$ is a non-zero polynomial of degree four, having local extreme points at $x= –1, 0, 1;$ then the set $S=\{x\in R :f\left(x\right)=f\left(0\right)\}$ contains exactly

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If at $x=1,$ the function $x^4-62x^2+ax+9$ attains its local maximum value, on the interval $\left[0,2\right]$, then the value of $a$ is

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Let $k$ and $K$ be the minimum and the maximum values of the function $f\left(x\right)=\frac{{\left(1+x\right)}^{0.6}}{1+x^{0.6}} \text{in} \left[0, 1\right]$, respectively, then the ordered pair $(k, K)$ is equal to:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Let $M$ and $m$ be respectively the absolute maximum and the absolute minimum values of the function, $f\left(x\right)=2x^3-9x^2+12x+5$ in the interval $[0,3]$ . Then $M-m$ is equal to

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

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

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The least value of $\alpha \in R$ for which, $4\alpha x^2+\frac{1}{x} \ge 1,$ for all $x>0,$ is 

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The maximum value of $f\left(x\right)=\frac{\log x}{x} (x\ne 0,x\ne 1)$ is

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The difference between the greatest and the least value of $f\left(x\right)=2\sin x+\sin 2x, x\in \left[0,\frac{3\pi }{2}\right]$ is

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The point on the curve $x^2=2y$ which is nearest to the point $\left(0,5\right)$ is

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If the function $f$ given by $f\left(x\right)=x^3-3\left(a-2\right)x^2+3ax+7$, for some $a\in R$ is increasing in $\left(0, 1\right]$ and decreasing in $\left[1, 5\right)$, then a root of the equation, $\frac{f\left(x\right)-14}{{\left(x-1\right)}^2}=0, \left(x\ne 1\right)$ is :