The difference between the maximum and minimum values of $f(x) = x^4e^{-x^2}$ for all $x \in R$ is:

Let $AP$ and $BQ$ be two vertical poles at points $A$ and $B$,respectively. If $AP=16 \, m, BQ=22 \, m$ and $AB=20 \, m,$ then find the distance of a point $R$ on $AB$ from the point $A$ such that $RP^2 + RQ^2$ is minimum. (in $, m$)

The maximum value of $(1/x)^x$ is -

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 $-4 < a < 4$

If $x$ and $y$ are two positive real numbers such that $xy = 4$,then the minimum value of $\left(\sqrt{x} + \frac{y^2}{2}\right)$ is

$A$ wire of length $20 \text{ cm}$ is bent in the form of a sector of a circle. The maximum area that can be enclosed by the wire is (in $\text{ cm}^2$)