$A$ wire of length $20 \ m$ is to be cut into two pieces. $A$ piece of length $\ell_1$ is bent to make a square of area $A_1$ and the other piece of length $\ell_2$ is made into a circle of area $A_2$. If $2A_1 + 3A_2$ is minimum,then $(\pi \ell_1) : \ell_2$ is equal to:

The sum of the absolute maximum and minimum values of the function $f(x) = |x^2 - 5x + 6| - 3x + 2$ in the interval $[-1, 3]$ is equal to:

If $xy = c^2$,what is the minimum value of $ax + by$ (where $a > 0, b > 0$)?

Let $f(x) = \begin{cases} |x| & \text{for } 0 < |x| \leqslant 2 \\ 1 & \text{for } x = 0 \end{cases}$. What is the nature of $f$ at $x = 0$?

For all real $x$,the minimum value of the function $f(x)=\frac{1-x+x^2}{1+x+x^2}$ is

$A$ tank with a rectangular base and rectangular sides, open at the top is made. The depth of the tank is $4 \text{ m}$ and its volume is $36 \text{ m}^3$. The cost of the base material is $\text{Rs. } 100 \text{ per m}^2$ and the cost of the side material is $\text{Rs. } 50 \text{ per m}^2$. Find the minimum cost of the tank.