The function $f(x) = x + \frac{a^2}{x}$, $a > 0$, $x \neq 0$ has a local maxima at

The maximum value of $f(x) = \frac{1}{4x^2 + 2x + 1}$ is

For the function $f(x) = \frac{x^4}{4} - 2x^3 + \frac{11}{2}x^2 - 6x$, which of the following statements are TRUE?

(A) Critical points of $f(x)$ are $x = 1, 2$ and $3$.

(B) $f(x)$ is increasing over $(1,2)$.

(C) $f(x)$ is decreasing over $(2,3)$.

(D) $f(x)$ is increasing over $(3,\infty)$.

Choose the correct answer from the options given below:

Let $f(x) = \log_e(\sin x), x \in (0, \pi)$, then which of the following statements is/are TRUE?

(A) $f(x)$ is increasing on $(0, \pi/2)$

(B) $f(x)$ is decreasing on $(\pi/2, \pi)$

(C) $f(x)$ is increasing on $(0, \pi)$

(D) $f(x)$ is decreasing on $(0, \pi)$

Choose the correct answer from the options given below:

If $x$ is real, then the minimum value of $x^2 - 8x + 25$ is

The local minimum value of $f(x) = x^3 + \frac{243}{x}$ is

Minimum value of D is :