Question

Match the entries in List I with entries in List II

Answer 1

Element A:
$f\left(x\right)=\frac{\tan x+\cot x}{2}-\left|\frac{\tan x-\cot x}{2}\right|$
$f\left(x\right)=\{\begin{array}{c}\cot x,\tan x\ge \cot x\\ \tan x,\tan x<\cot x\end{array}$
There are $4$ points $(\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4})$ where the above function is continuous but not differentiable in $(0,2\pi )$
Element B:
$f\left(x\right)=min\{1,1+x^3,x^2-3x+3\}$
$f\left(x\right)=\{\begin{array}{c}1+x^3,x\le 0\\ 1,0<x\le 1\\ x^2-3x+3,1<x\le 2\\ 1,x>2\end{array}$
$f^{\prime }\left(x\right)=\{\begin{array}{c}3x^2,x\le 0\\ 0,0<x\le 1\\ 2x-3,1<x\le 2\\ 1,x>2\end{array}$
There are $2$ point $x=1$ and $x=2$ where the above function is not differentiable.
Element C:
$f\left(x\right)={(x+4)}^{1/3}$
$f^{\prime }\left(x\right)=\frac{1}{3}{(x+4)}^{2/3}$
Not derivable at $x=-4$
Element D:
$f\left(x\right)=\{\begin{array}{c}-\frac{\pi }{2}\ln \left(\frac{x.2}{\pi }\right)+\frac{\pi }{2},0<x\le \frac{\pi }{2}\\ \pi -x,\frac{\pi }{2}<x<\frac{3\pi }{2}\end{array}$
$f^{\prime }\left(x\right)=\{\begin{array}{c}-\frac{\pi }{2x}0<x<\frac{\pi }{2}\\ -1\frac{\pi }{2}<x<\frac{3\pi }{2}\end{array}$
$f^{\prime }\left(\frac{{\pi }^{-}}{2}\right)=f^{\prime }\left(\frac{{\pi }^{+}}{2}\right)=-1$
function differentiable at $x=\frac{\pi }{2}$