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

A wire of length $l$ is cut into two parts. One part is bent into a circle and other into a square. Show that the sum of areas of the circle and square is the least, if the radius of circle is half the side of the square

If $f\left(x\right)=\cos \left|x\right|-2ax+b$ is a function, which increases for all $x,$ then the maximum value of $2a+1$ is equal to

Find the points of inflection of the function $f\left(x\right)=2x^3+6x^2-5x+1$.

Find the points of inflection of the function $f\left(x\right)=x^3-3x^2-1$.

Find the points of inflection of the curve $y=x^3-3x^2-5x+7$.