If $f\left(x\right)=x^{\alpha }\log x$ and $f\left(0\right)=0$, then the value of $\alpha$ for which Rolle's theorem can be applied in $\left[0, 1\right]$ is

Let, $f\left(x\right)=\sin x$ and $g\left(x\right)=\left\{\begin{array}{ll}\max \left\{f\right(t),& 0\le t\le x , 0\le \mathrm{x}\le \mathrm{\pi }\}\\ \frac{1-\cos x}{2},& x>\mathrm{\pi }\end{array}\right.$. Then $g\left(x\right)$ is

Let $f:R\to R$ be a function defined as $f\left(x\right)=\left\{\begin{array}{ccc}3\left(1-\frac{\left|x\right|}{2}\right)& \text{if}& \left|x\right|\le 2\\ 0& \text{if}& \left|x\right|>2\end{array}\right..$

Let $g:R\to R$ be given by $g\left(x\right)=f(x+2)-f(x-2).$ If $n$ and $m$ denote the number of points in $R$ where $g$ is not continuous and not differentiable, respectively, then $n+m$ is equal to ________.

A function $f$ is defined on $[-3,3]$ as

$f\left(x\right)=\left\{\begin{array}{c}\min \left\{\left|x\right|,2-x^2\right\},-2\le x\le 2\\ \left[\right|x\left|\right] , 2<\left|x\right|\le 3\end{array}\right.$

where $\left[x\right]$ denotes the greatest integer $\le x$. The number of points, where $f$ is not differentiable in $\left(-3,3\right)$ is ___ .