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

The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ is:

If $x=1$ and $x=2$ are extreme points of $f(x)=\alpha \log x+\beta x^2+x$,where $\alpha$ and $\beta$ are constants,then the value of $\alpha^2+2 \beta$ is

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function $f(x) = 9x^4 + 12x^3 - 36x^2 + 25, x \in R$,then

Find the local minimum value of the function $f$ given by $f(x) = 3 + |x|, x \in R$.

Statement-$I$: The sequence $a_n = \frac{n^2}{n^3 + 200}, n \in N$ has its $7^{th}$ term as the largest term.
Statement-$II$: The function $f(x) = \frac{x^2}{x^3 + 200}$ attains a local maximum at $x = 7$.