Maxima and Minima

Find the area of the right angled triangle of least area that can be drawn so as to circumscribe a rectangle of sides '$a$' and '$b$', the right angle of the triangle coinciding with one of the angles of the rectangle.

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 $f\left(x\right).$

Choose the height of the cone of maximum volume that can be inscribed in a sphere of radius $12 \mathrm{cm}$:

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?

In the interval $\left[-2,\frac{9}{2}\right]$, the absolute maximum value of the function $f\left(x\right)=4x-\frac{1}{2}{x}^2$ is equal to

Of all the closed right circular cylindrical cans with volume $54 \pi {\mathrm{cm}}^3,$ the height of the can that has the maximum surface area is

The absolute maximum of the function $f:[0,4]\to \mathrm{ℝ}$ defined by $f\left(x\right)=x\sqrt{16-x^2}$, that is, $\max \left\{f\right(x):x\in [0,4\left]\right\}$, is

The function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)=\frac{1}{x^4-2x^2+7}$ has a local minimum at

The least value of $f\left(x\right)=\frac{x^3}{3}-abx$ occurs at $x=$

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 $\left(0,\frac{\mathrm{\pi }}{2}\right)$