The function $f(x)=2|x|+|x+2|-||x+2|-2|x||$ has a local minimum or a local maximum at $x=$

If the set of all values of $a$,for which the equation $5x^3 - 15x - a = 0$ has three distinct real roots,is the interval $(\alpha, \beta)$,then $\beta - 2\alpha$ is equal to . . . . . . .

Find the maximum value of $(1/x)^x$.

Suppose $f(x)$ is a polynomial of degree four,having critical points at $-1, 0, 1$. If $T = \{x \in \mathbb{R} \mid f(x) = f(0)\}$,then the sum of squares of all the elements of $T$ is

What is the maximum area of a rectangle that can be formed with a fixed perimeter $p \ cm$?

Let $x_0$ be the point of local minima of $f(x) = \overline{a} \cdot (\overline{b} \times \overline{c})$ where $\overline{a} = x \hat{i} - 2 \hat{j} + 3 \hat{k}$,$\overline{b} = -2 \hat{i} + x \hat{j} - \hat{k}$,and $\overline{c} = 7 \hat{i} - 2 \hat{j} + x \hat{k}$. Then the value of $\overline{a} \cdot \overline{b}$ at $x = x_0$ is: