If $m$ and $M$ are respectively the absolute minimum and absolute maximum values of a function $f(x)=2x^3+9x^2+12x+1$ defined on $[-3,0]$, then $m+M=$

If $P$ is a point on the segment $AB$ of length $12 \text{ cm}$,then the position of $P$ for $AP^{2} + BP^{2}$ to be minimum is such that

If the function $y = a \log x + bx^2 + x$ has extreme values at $x = 1$ and $x = 2$,then $(a, b) = \dots$

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ be defined by $f(x)=\begin{cases} \frac{6x+\sin x}{2x+\sin x} & \text{if } x \neq 0 \\ \frac{7}{3} & \text{if } x=0 \end{cases}$. Then which of the following statements is (are) True?
$(A)$ The point $x=0$ is a point of local maxima of $f$
$(B)$ The point $x=0$ is a point of local minima of $f$
$(C)$ Number of points of local maxima of $f$ in the interval $[\pi, 6\pi]$ is $3$
$(D)$ Number of points of local minima of $f$ in the interval $[2\pi, 4\pi]$ is $1$

The absolute minimum value of the function $f(x) = x^3 - 18x^2 + 96x$ for $x \in [0, 9]$ is:

The function $f(x) = \frac{x}{1 + x \tan x}$ attains its maximum value at: