The sufficient conditions for the function $f:R \to R$ to have a local maximum at $x = a$ are:

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:

Find two positive numbers whose sum is $16$ and the sum of whose cubes is minimum.

The maximum slope of the curve $y=-x^{3}+3x^{2}+2x-27$ is

The maximum area of a rectangle with a perimeter of $176 \ cm$ is .......... $sq. \ cm$.

Find the points at which the function $f$ given by $f(x)=(x-2)^{4}(x+1)^{3}$ has
$(i)$ local maxima
$(ii)$ local minima
$(iii)$ point of inflexion