Let $f(x)=(x+3)^2(x-2)^3$ for $x \in [-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$ respectively in $[-4,4]$,then the value of $M-m$ is:

Statement-$I$: Let the function $f(x) = \begin{cases} -\frac{x}{2} & x < 0 \\ 7x + 8 & x \geq 0 \end{cases}$. Then $f(x)$ has a local minimum at $x = 0$.
Statement-$II$: If $f(a) < f(a - h)$ and $f(a) < f(a + h)$ for a sufficiently small $h > 0$,then $f(x)$ has a local minimum at $x = a$.

An open-topped box is to be constructed by removing equal squares from each corner of a $3 \text{ m}$ by $8 \text{ m}$ rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.

Find the local maximum and local minimum values of $f(x) = \sin 2x - x$ on the interval $[0, \pi]$.

The condition for the function $f(x) = x^3 + px^2 + qx + r$ $(x \in R)$ to have no extreme value is:

The point in the interval $[0, 2\pi]$, where $f(x) = e^x \sin x$ has maximum slope, is