The function $f(x) = x^3 - 6x^2 + 9x + 2$ has a maximum value when $x$ is

The shortest distance of the point $(0,0)$ from the curve $y = e^x + e^{-x}$ is-

The two positive numbers with sum $t$,and the sum of their squares is minimum are

The function $f(x) = x^2 \log x$ in the interval $(1, e)$ has

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:

The maximum value of ${\left( {\frac{1}{x}} \right)^{2{x^2}}}$ is