An extreme value of $f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}$ in $(0, \pi / 2)$ 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 $-4 < a < 4$

Let $f(x)$ be a polynomial of degree $5$ such that $x=\pm 1$ are its critical points. If $\mathop {\lim }\limits_{x \to 0} \left(2+\frac{f(x)}{x^{3}}\right)=4,$ then which one of the following is not true?

Local maximum and local minimum values respectively of the function $f(x)=(x-1)(x+2)^2$ are

Let the radius and height of a right circular cylinder be related as $r^2 + h = 6$. If the volume of the cylinder is maximum,then the value of $\frac{r}{h}$ is:

The maximum area of a rectangle that can be formed with a fixed perimeter of $20$ units is $ . . . . . . $ sq units.