Critical Points

The number of critical points of $f\left(x\right)=\max \left\{\sin x, \cos x\right\}, \forall x\in (-2\mathrm{\pi }, 2\mathrm{\pi })$ is

Let $f\left(x\right)=\frac{x-1}{x^2}$, then which of the following is/are correct?

The function $f\left(x\right)=\left\{\begin{array}{ccc}x+2& ,& x<-1\\ x^2& ,& -1\le x<1\\ (x-2{)}^2& ,& x\ge 1\end{array}\right.$

The function $\sqrt{a{x}^3+b{x}^2+cx+d}$ has its non-zero local minimum and maximum values at $-2$ and $2$ respectively. If' ‘ $a$' is root of $x^2-x-6=0$, find the possible values of $a, b, c$ and $d$.

Let $f\left(x\right)=x^3-x^2+x+1$ and $g\left(x\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\begin{array}{l}f\left(t\right): 0\le t\le x; 0\le x\le 1\\ \begin{array}{cc}3-x& ;1<x\le 2\end{array}\end{array}\right.\begin{array}{}\end{array}$. Discuss the continuity and differentiability of $g\left(x\right)$ in the interval $\left(0, 2\right)$.

For the following questions, choose the correct answers from the codes (a), (b), (c) and (d) defined as follows:
Let $f\left(0\right)=0,f\left(\frac{\pi }{2}\right)=1,f\left(\frac{3\pi }{2}\right)=-1$ be a continuous and twice differentiable function.
Statement I: $\left|f^{\text{'}}\left(x\right)\right|\le 1$ for atleast one $x\in \left(0,\frac{3\pi }{2}\right)$ because
Statement II: According to Rolle's theorem, if $y=g\left(x\right)$ is continuous and differentiable $\forall x\in \left[a,b\right]$ and $g\left(a\right)=g\left(b\right)$, then there exists atleast one $c$ such that $g^{\text{'}}\left(c\right)=0$.

If $f\left(x\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left(\mathrm{s}\mathrm{i}\mathrm{n}x, \mathrm{c}\mathrm{o}\mathrm{s}x\right),\forall x\in R$. Then, find the number of critical points lying in the interval $\left(-2\mathrm{\pi }, 2\mathrm{\pi }\right)$.