The minimum value of $4e^{2x} + 9e^{-2x}$ is:

If $x$ and $y$ are two variables such that $x > 0$ and $xy = 1$,what is the minimum value of $x + y$?

The minimum value of the function $f(x) = x \log x$ is

Let $f: D \rightarrow R$, $D \subseteq R$, $c \in D$ and $r$ be a non-zero real number. Consider the following statements:
$Y$. $c$ is an extreme point of $f \Rightarrow c$ is an extreme point of $rf$
$M$. $c$ is an extreme point of $f \Rightarrow c$ is an extreme point of $r+f$
Which of the following is correct?

The range of $a \in R$ for which the function $f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)$ where $x \neq 2 n \pi, n \in N$,has critical points,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.