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 $, m$)

If $f(x)$ is a function such that $f^{\prime}(x)=(x-1)^{2}(4-x),$ then

If $f(x) = \int\limits_0^x {{e^{\frac{{ - {t^2}}}{2}}}} \left( {1 - {t^2}} \right)\,dt$,then $f(x)$ is minimum at $x = \dots$

The maximum value of the function $f(x)=2x^3-15x^2+36x-48$ on the set $A=\{x | x^2+20 \leq 9x\}$ is

For the function $f(x) = x \cos \frac{1}{x}, \quad x \geq 1$,consider the following statements:
$(A)$ For at least one $x$ in the interval $[1, \infty), f(x+2)-f(x) < 2$
$(B)$ $\lim _{x \rightarrow \infty} f^{\prime}(x) = 1$
$(C)$ For all $x$ in the interval $[1, \infty), f(x+2)-f(x) > 2$
$(D)$ $f^{\prime}(x)$ is strictly decreasing in the interval $[1, \infty)$
Which of the following combinations of statements is correct?

Find two positive numbers $x$ and $y$ such that their sum is $35$ and the product $x^{2} y^{5}$ is a maximum.