Find the point where $f(x) = 2\sin x + \cos 2x$ has a maximum value in the interval $0 \leq x \leq 2\pi$.

The sum of two non-zero numbers is $4$. The minimum value of the sum of their reciprocals is

Let $\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac{\pi}{6}\right)$. Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by $g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}$. Then,which of the following statements is/are $TRUE$?
$(A)$ The minimum value of $g(x)$ is $2^{7/6}$
$(B)$ The maximum value of $g(x)$ is $1 + 2^{1/3}$
$(C)$ The function $g(x)$ attains its maximum at more than one point
$(D)$ The function $g(x)$ attains its minimum at more than one point

The maximum value of $2x^3 - 24x + 107$ in the interval $[-3, 3]$ is

If rectangles are inscribed in a circle of radius $r$ units,then the dimensions of the rectangle which has maximum area are:

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $R = \sqrt{3}$ is