The number $10$ is divided into two parts such that the sum of double of the first part and the square of the second part is minimum. The two parts are respectively:

Let $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^3+ax^2+b \ln|x|+1, x \neq 0$. Let $m$ and $M$ respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2, -\frac{1}{2}\right]$. Then $|M+m|$ is equal to (Take $\ln 2 \approx 0.7$):

The sum of two numbers is $20$. If the product of the square of one number and the cube of the other is maximum, then the numbers are:

What is the maximum area of a rectangle that can be formed with a fixed perimeter $p \ cm$?

If an open cylinder of given surface area has maximum volume,then its radius is

The minimum value of the function $f(x) = x \log x$ is