The number $28$ is divided into two positive parts such that the sum of the cube of one part and the square of the other part is minimum. Find the absolute difference between the two parts.

Let $a \in R$ and let $f: R \rightarrow R$ be given by $f(x)=x^5-5x+a$. Then
$(A)$ $f(x)$ has three real roots if $a > 4$
$(B)$ $f(x)$ has only one real root if $a > 4$
$(C)$ $f(x)$ has three real roots if $a < -4$
$(D)$ $f(x)$ has three real roots if $-4 < a < 4$

For the function $f(x)=x^3-6x^2-12x-3$,$x=2$ is a

There are $50$ apple trees in an orchard. Each tree produces $800$ apples. For each additional tree planted in the orchard,the output per tree drops by $10$ apples. The number of trees that should be added to the existing orchard to maximize the total output is:

If $f(x)=x^5-5 x^4+5 x^3-10$ has its local maxima and minima at $x=a$ and $x=b$ respectively,then $2 a+b$ is equal to

The function,$f(x)=x \sqrt{1-x}$,where $x \in(0,1)$,has a local maximum at $x=$