If $f(x) = 7e^{\sin^2 x} - e^{\cos^2 x} + 2$,then $\sqrt{7f_{\min} + f_{\max}}$ is equal to

Let $M$ and $m$ be respectively the local maximum and the local minimum values of the function $f(x) = 2x^3 - 9x^2 + 12x + 5$ in the interval $[0, 3]$. Then $M - m$ is equal to

The set of all values of $a$ for which the function $f(x) = (a^2 - 3a + 2) \left( \cos^2 \frac{x}{4} - \sin^2 \frac{x}{4} \right) + (a - 1)x + \sin 1$ does not possess critical points is

$A$ rectangle with its sides parallel to the $X$-axis and $Y$-axis is inscribed in the region bounded by the curves $y=x^2-4$ and $y=\frac{4-x^2}{2}$. The maximum possible area of such a rectangle is closest to the integer.

If the function $f(x)=2x^{3}-9ax^{2}+12a^{2}x+1$ where $a>0$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2}=q$,then $a$ is equal to

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?