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.

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 $f\left(x\right).$

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?

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 $f:\left(0,1\right)\to \mathrm{ℝ}$ be the function defined as $f\left(x\right)=\left[4x\right]{\left(x-\frac{1}{4}\right)}^2\left(x-\frac{1}{2}\right)$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$. Then which of the following is(are) true?

Find the maximum value of $f\left(x\right)=\sin x$.

Let $f:R\to R$ and graph of function $y=f^{\text{'}}\left(x\right)$ is shown below:

Let

$a=$Number of point of inflection of $y=f\left(x\right)$

$b=$Number of point local minima of $y=f\left(x\right)$

$c=$Number of point local maxima of $y=f\left(x\right)$

$d=$Number of critical point where  neither maxima nor minima occurs

Then, the value of $\frac{a+b+d}{a-c}$ is

For the function $f\left(x\right)=2^x-1-x^2$, which of the following statement(s) are true?