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.

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 $f\left(x\right).$

A sector is removed from a metallic disc and the remaining region is bent into the shape of a circular conical funnel with volume $2\sqrt{3}\pi$. The least possible diameter of the disc is

Find local maxima of $\tan 4x$ at critical point in $(0,\frac{\pi }{4})$

Find local maxima of $\tan 3x$ at critical point in $\left(0,\frac{\mathrm{\pi }}{2}\right)$

Find local maxima of $\tan 2x$ at critical point in $\left(0,\frac{\mathrm{\pi }}{2}\right)$

Find local maxima and minima of $\cot x$ at critical point in $\left[0,\mathrm{\pi }\right]$

Find local maxima of $\tan x$ at critical point in $\left(0,\mathrm{\pi }\right)$

The maximum value of $y=x(\log x{)}^2$ is _____

Find local maxima and minima of $\cot x$ at critical point in $\left(0,\mathrm{\pi }\right)$

For all values of $\theta \in \left(0,\frac{\mathrm{\pi }}{2}\right)$, the determinant of the matrix $\left[\begin{array}{ccc}-2& \tan \theta +{\sec }^2\theta & 3\\ -\sin \theta & \cos \theta & \sin \theta \\ -3& -4& 3\end{array}\right]$ always lies in the interval

Let $p\left(x\right)$ be a real polynomial of degree $4$ having extreme values at $x=1$ and $x=2.$ If $\underset{x\to 0}{\lim }\frac{p\left(x\right)}{x^2}=1$, then $p\left(4\right)$ is equal to

The maximum value of $f\left(x\right)=2\sin x+\sin 2x$ in the interval $\left[0, \frac{3\pi }{2}\right]$, is

The number of values of x, where the function  $f\left(x\right)= \cos x+\text{cos}\left(\sqrt{2}\text{x}\right)$  attains its maximum, is 

The number of values of $x,$ where $f\left(x\right)=\cos x+\cos \sqrt{2}x$ attains its maximum is