Maxima and Minima

The absolute minimum value of the function $f\left(x\right)=2\sin x$ in $\left[0,\frac{3\mathrm{\pi }}{2}\right]$ is $?$

If $x>0,$ then find the greatest value of the expression $\frac{x^{100}}{1+x+x^2+x^3+\dots ..+x^{200}}.$

Find the least value of $4x+\frac{16}{x}$.

Prove that $f\left(x\right)=\cos x-1+\frac{x^2}{2!}-\frac{x^4}{4!}$ has its maximum value at $x=0.$

Show that the rectangle of maximum perimeter which can be inscribed in a circle of radius $10$ cm is a square of side $10\sqrt{2}$ cm.

Water is being filled at the rate of 100 litre/min in a conical vessel (vertex downwards) with axis vertical. Its semi vertical angle is ${\tan }^{-1}\frac{5}{12}$. The rate (in square meter per minute) at which the wet curved surface area of the tank is increasing, when the depth of water in tank is 3 meters is

The minimum value of ${\tan }^4\theta +{\cot }^4\theta$ is less than or equal to$?$

Divide $100$ into two parts so that sum of their squares is minimum.

Raj is playing with a spring by throwing it in the air which is moving along the function $f\left(x\right)=3x^4-4x^3-12x^2+5$.

Find the critical points of $x$ when it touches the $x-$axis.

Find the maximum and minimum for 

$a. y=4x$

$b. y=x^2-4x$

$c. y=6x-x^2$

The maximum value of $x^2{e}^{-x^2}$ is equal to 

What is maxima and minima?

If $a,b>0$ such that $a^3+b^3=2$ then the maximum value of $a+b$ is $?$

Graph of $y=f\left(x\right)$ is given, then 

The height $h$ in meters of an object varies with time $\text{'}t\text{'}$ in seconds as $h=10t-5t^2$. Then the maximum $(in m)$ height attained by the object is?

Manufacturer can sell $x$ items at a price of rupees $\left(5-\frac{x}{100}\right)$ each. The cost price of $x$ items is $\left(\frac{x}{5}+500\right)$. Find the number of items he should sell to earn maximum profit.

An open topped box is to be constructed by removing equal squares from each corner of a $3$ metre by $8$ metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.

An Apache helicopter of enemy is flying along the curve given by $y=x^2+7$. A soldier, placed at $\left(3, 7\right)$, wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.

Find the absolute maximum and minimum values of a function $\mathrm{f}$ given by

$\mathrm{f}\left(\mathrm{x}\right)=2{\mathrm{x}}^3-15{\mathrm{x}}^2+36\mathrm{x}+1$ on the interval $\left[1, 5\right]$.

Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.