The function $f(x) = x^5 - 5x^4 + 5x^3 - 10$ has a local maximum when $x$ is equal to:

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:

If $x+y=k, x>0, y>0$,then $x^2+y^2$ is minimum,if

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f(x) = \int_{0}^{x^{2}} \frac{t^{2}-5t+4}{2+e^{t}} dt$,then the ordered pair $(m, n)$ is equal to

The function $f(x) = \int_{-1}^x {t({e^t} - 1)(t - 1){(t - 2)}^3{(t - 3)}^5} dt$ has a local minimum at $x = $ ..........

If the absolute maximum and absolute minimum values of the function $f(x) = x^3 - 2x^2 + x - 3$ defined on $[0, 2]$ are $M$ and $m$ respectively, then $M + m =$