Critical Point

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.

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).

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

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

$A\left(5,-3\right), C\left(7,8\right)$ 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

Let a function $f\left(x\right)$ be continuous in an interval $\left[a,b\right]$. Let $\delta >0$ be a very small real number. Let $\mathrm{c}\in \left(a,b\right)$ be such that $f\left(\mathrm{c}-\delta \right)<f\left(\mathrm{c}\right)$ and $f\left(c+\delta \right)<f\left(c\right)$ for every $\delta >0$. Let $\left(f\left(\alpha -\delta \right)-f\left(\alpha \right)\right)\left(f\left(\alpha +\delta \right)-f\left(\alpha \right)\right)<0 \forall \alpha \in \left(a,b\right)$ and $\alpha \ne c$. Then

Let $f\left(x\right)=\frac{6x^2-18x+21}{6x^2-18x+17}$. If $m$ is the maximum value of $f\left(x\right)$ and $f\left(x\right)>n\forall x\in \mathrm{ℝ}$ Then $14 \mathrm{m}-7 \mathrm{n}=$

If $x,y,z>0$ and $x+y+z=1$, then the least value of $\frac{5x}{2-x}+\frac{5y}{2-y}+\frac{5z}{2-z}$ is?

The function $f\left(x\right)=x^2\ln \left(3x\right)+6$ has

Let $f\left(x\right)=\left\{\begin{array}{lc}4x-x^3+\ln \left(b^2-3b+3\right),& 2\le x<3\\ x-18& x\ge 3\end{array}\right.$. Find all the possible real values of $b$ such that $f\left(x\right)$ has the smallest value at $x=3$.

Let $f\left(x\right)=\frac{x^2+x-1}{x^2-x+1}$, then the largest value of $f\left(x\right)\forall x\in \left[-1,3\right]$ is

Graph of $y=f\left(x\right)$ is given, then 

The height $h$ in meters of an object varies with time $\text{'}t\text{'}$ in seconds as $h=10t-5t^2$. Then the maximum $(in m)$ height attained by the object is?

Maximum value of the function $f\left(x\right)=\left|\sqrt{x^2+4x+5}-\sqrt{x^2+2x+5}\right|$ is 

Consider the function $f:(-\infty ,\infty )\to (-\infty ,\infty )$ defined by $f\left(x\right)=\frac{x^2-a}{x^2+a},a>0$. Which of the following is not true?