The maximum value of $f(x) = \sin (x)$ in the interval $[-\pi / 2, \pi / 2]$ is

The maximum area of a rectangle that can be inscribed in a circle of radius $2 \text{ unit}$ is (in square unit)

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ be defined by $f(x)=\begin{cases} \frac{6x+\sin x}{2x+\sin x} & \text{if } x \neq 0 \\ \frac{7}{3} & \text{if } x=0 \end{cases}$. Then which of the following statements is (are) True?
$(A)$ The point $x=0$ is a point of local maxima of $f$
$(B)$ The point $x=0$ is a point of local minima of $f$
$(C)$ Number of points of local maxima of $f$ in the interval $[\pi, 6\pi]$ is $3$
$(D)$ Number of points of local minima of $f$ in the interval $[2\pi, 4\pi]$ is $1$

The absolute minimum value of $x^4-x^2-2x+5$ is

Find two positive numbers $x$ and $y$ such that $x+y=60$ and $x y^{3}$ is maximum.

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