Maxima and Minima

If $P\left(x\right)$ be a polynomial of degree $3$ satisfying $P\left(-1\right)=10, P\left(1\right)=-6$ and $P\left(x\right)$ has maximum at $x=-1$ and $P^{\text{'}}\left(x\right)$ has minima at $x=1$. Find the distance between the local maximum and local minimum of the curve.

Find a point on the curve ${\text{x}}^2+2{\text{y}}^2=6$ whose distance from the line $\text{x}+\text{y}=7$, is minimum.

The plan view of a swimming pool consists of a semicircle of radius $r$ attached to a rectangle of length $2r$ and width $s$. If the surface area $A$ of the pool is fixed, for what value of $r$ and $s$ the perimeter $P$ of the pool is minimum.

Suppose f(x) real valued polynomial function of degree 6 satisfying the following conditions;

(a) f has minimum value at x = 0 and 2

(b) f has maximum value at x = 1

(c) For all $\text{x, }\underset{\text{x}\to 0}{\text{limit}}\frac{1}{\text{x}}\ell \text{n}\left|\begin{array}{ccc}\frac{\text{f}\left(\text{x}\right)}{\text{x}}& 1& 0\\ 0& \frac{1}{\text{x}}& 1\\ 1& 0& \frac{1}{\text{x}}\end{array}\right|=2$.

Determine f(x).

Investigate for maxima & minima for the function, $\text{f}\left(\text{x}\right)$, $\text{f}\left(\text{x}\right)=\underset{1}{\overset{\text{x}}{\int }}\left[2\left(\text{t}-1\right){\left(\text{t}-2\right)}^3+3{\left(\text{t}-1\right)}^2{\left(\text{t}-2\right)}^2\right]\text{dt}$

A cubic $f\left(x\right)$ vanishes at $x=-2$ & has relative minimum/maximum at $x=-1, x=\frac{1}{3}$.
Find $\underset{-1}{\overset{1}{\int }}f\left(x\right)dx$ , if coefficient of $x^3=1$ in $f\left(x\right).$

An open box with a square base is to be made out of a given quantity of cardboard of area $c^2$ square units. What would be the maximum volume of the box?

What would be the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone?

A window is in the form of a rectangle with length $L$ and breadth $B$ surmounted by a semi-circle. If the total perimeter of the window is $30 \mathrm{m}$, then the dimensions of the window so that maximum light is admitted would be

Let $P$ be the median, $Q$ be the mean and $R$ be the mode of observations $x_1, x_2, x_3,\dots x_n$. Let $S=\sum _{i=1}^n{\left(2x_i-a\right)}^2$. $S$ takes minimum value, when $a$ is equal to

$\underset{0\le x\le \pi }{\max }\left\{x-2\sin x\cos x+\frac{1}{3}\sin 3x\right\}=$

If the total maximum value of the function $f\left(x\right)={\left(\frac{\sqrt{3e}}{2\sin x}\right)}^{{\sin }^{2}x}, x\in \left(0, \frac{\pi }{2}\right),$ is $\frac{k}{e},$ then ${\left(\frac{k}{e}\right)}^8+\frac{k^8}{e^5}+k^8$ is equal to

If $a_{\alpha }$ is the greatest term in the sequence $a_n=\frac{n^3}{n^4+147}, n=1, 2, 3....,$ then $\alpha$ is equal to $\_\_\_\_\_\_$

A square piece of tin of side $30 \mathrm{cm}$ is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in ${\mathrm{cm}}^2$) is equal to

$A\left(5,-3\right), C\left(7,8\right)$ and $B\left(t,0\right)$, $0\le t\le 4$. The perimeter is maximum at $t=\alpha$ and minimum at $t=\beta$, then ${\alpha }^2+{\beta }^2$ is

Maximum value of $f\left(x\right)=x-\sin 2x+\frac{1}{3}\sin 3x$ in $\left[0,\mathrm{\pi }\right]$ is