The sum of all the local minimum values of the twice differentiable function $f:R\to R$ defined by $f\left(x\right)=x^3-3x^2-\frac{3f^{"}\left(2\right)}{2}x+f^{"}\left(1\right)$ is:

Let a function $g:\left[0,4\right]\to R$ be defined as

$g\left(x\right)=\left\{\begin{array}{cc}\underset{0\le t\le x}{\max \left\{t^3-6t^2+9t-3\right\}},& 0\le x\le 3\\ 4-x,& 3<x\le 4\end{array}\right.$

then the number of points in the interval $\left(0, 4\right)$ where $g\left(x\right)$ is NOT differentiable, is _________.

Let $f:R\to R$ be defined as $f\left(x\right)=\left\{\begin{array}{cc}-\frac{4}{3}{x}^3+2x^2+3x,& x>0\\ 3x{e}^x ,& x\le 0\end{array}.\right.$ Then $f$ is increasing function in the interval

If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is _____.

Let $f$ be a real valued function, defined on$R-\left\{-1,1\right\}$ and given by $f\left(x\right)=3{\log }_{\mathrm{e}}\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}$. Then in which of the following intervals, function $f\left(x\right)$ is increasing?