If the function $f(x) = \frac{x}{5} + \frac{5}{x}, (x \neq 0)$ attains its relative maximum value at $x = a$, then $\sqrt{a^2 + 2a - 6} = $

For which values of $x$ is the function $f(x) = \sin x + \cos 2x$ $(x > 0)$ minimum?

$A$ wire of length $22 \; m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then,the length of the side of the equilateral triangle,so that the combined area of the square and the equilateral triangle is minimum,is

Let a function $f(x)$ be defined as $f(x) = \begin{cases} \cos^{-1}(\mu) + x^2, & 0 < x < 1 \\ 4x, & x \geqslant 1 \end{cases}$. The function $f(x)$ can have a local minimum at $x = 1$ if the value of $\mu$ lies in the interval:

If $f(x) = 2x^3 - 3x^2 - 12x + 5$ and $x \in [-2, 4]$,then the maximum value of the function occurs at which value of $x$?

The triangle of maximum area that can be inscribed in a given circle of radius $r$ is ...... .