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

If $A = \{x \in R : \frac{\pi}{4} \leq x \leq \frac{\pi}{3}\}$ and $f(x) = \sin x - x$,then $f(A)$ is equal to

On the interval $[0,1]$,the function $f(x) = x^{25}(1-x)^{75}$ takes its maximum value at the point

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta > 0$ be a very small real number. Let $c \in (a, b)$ be such that $f(c - \delta) < f(c)$ and $f(c + \delta) < f(c)$ for every $\delta > 0$. Let $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) < 0$ for all $\alpha \in (a, b)$ and $\alpha \neq c$. Then:

The minimum value of the function $f(x) = x^{10} + x^2 + \frac{1}{x^{12}} + \frac{1}{1 + \sec^{-1} x}$ is

For all $x \in \mathbb{R}$, the minimum value $\frac{1}{3}$ and the maximum value $3$ of $f(x) = \frac{x^2+x+1}{x^2-x+1}$ occur at $l$ and $m$ respectively. Then $l+m$ is equal to: