The real number $x$ when added to its inverse gives the minimum value of the sum at $x$ equal to

Let $\exp(x)$ denote the exponential function $e^x$. If $f(x) = \exp\left(x^{\frac{1}{x}}\right)$ for $x > 0$,then the minimum value of $f$ in the interval $[2, 5]$ is:

If the sum of two numbers is $3$,find the maximum value of the product of the first number and the square of the second number.

Find the local maximum and local minimum values of the function $f$ given by $f(x) = 3x^4 + 4x^3 - 12x^2 + 12$.

For every twice differentiable function $f : R \rightarrow [-2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ There exist $r, s \in R$,where $r < s$,such that $f$ is one-one on the open interval $(r, s)$.
$(B)$ There exists $x_0 \in (-4, 0)$ such that $|f'(x_0)| \leq 1$.
$(C)$ $\lim_{x \rightarrow \infty} f(x) = 1$.
$(D)$ There exists $a \in (-4, 4)$ such that $f(a) + f''(a) = 0$ and $f'(a) \neq 0$.

One maximum point of ${\sin ^p}x{\cos ^q}x$ is