Let $f(x)=3^{(x^{2}-2)^{3}+4}, x \in R$. Then which of the following statements are true?
$P: x=0$ is a point of local minima of $f$
$Q: x=\sqrt{2}$ is a point of inflection of $f$
$R: f^{\prime}$ is increasing for $x>\sqrt{2}$

Let the cubic polynomial $f(x) = x^3 - px + q$ have three real roots,where $p > 0$ and $q > 0$. Which of the following is true?

All the points on the curve $y^{2}=4a|x+a \sin(x/a)|$,where the tangent is parallel to the $x$-axis,lie on:

In the interval $[-2, 4]$,the absolute maximum of $f(x) = 2x^3 - 3x^2 - 12x + 5$ occurs at $x =$

The maximum value of $\left(\frac{1}{x}\right)^x$ is

$A$ rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$,the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are:
$(A)$ $24$
$(B)$ $32$
$(C)$ $45$
$(D)$ $60$