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

At what value of $x$ does the function $f(x) = x^x$ $(x > 0)$ attain its minimum value?

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 minimum distance of a point on the curve $y=x^2-4$ from the origin is

The minimum value of the function $y = 2x^3 - 21x^2 + 36x - 20$ is

Let $f: R \to R$,$f(x) = ax^3 + bx^2 + cx + d$ have no extreme value. Then which of the following is always correct?