The function $\sin x \left(1+\cos x\right)$ at $x=\frac{\pi }{3}$ , is

Important Questions on Application of Derivatives

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 :

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

EASY

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

The maximum value of the function $f\left(x\right)=2x^3-15x^2+36x+4$ is attained at

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 :

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 

HARD

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

Among all sectors of a fixed perimeter, choose the one with maximum area. Then the angle at the center of this sector (i.e., the angle between the bounding radii) is-

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

A wire of length $2$ units is cut into two parts which are bent respectively to form a square of side $=x$ units and a circle of radius $=r$ units. If the sum of the areas of the square and the circle so formed is minimum, then

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

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 

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:

MEDIUM

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

If non-zero real numbers $b$ and $c$ are such that  $min f\left(x\right)>max g\left(x\right)$, where $f\left(x\right)=x^2+2bx+2c^2$ and $g\left(x\right)=-x^2-2cx+b^2, \left(x\in R\right)$; then $\left|\frac{c}{b}\right|$ lies in the interval 

HARD

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

Let $f\left(x\right)=\alpha x^2-2+\frac{1}{x}$ where $\alpha$ is a real constant. The smallest $\alpha$ for which $f\left(x\right)\ge 0$ for all $x>0$ is-

HARD

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

Let $f\left(x\right)=x^2+\frac{1}{x^2}$ and $g\left(x\right)=x-\frac{1}{x}, x\in R-\left\{-1, 0, 1\right\}.$ If $h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ , then the local minimum value of $h\left(x\right)$ 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

The minimum distance of a point on the curve $y=x^2-4$ from the origin 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:

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:

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