Question

Let $f\left(x\right)=\{\begin{array}{cc}x+2,& 0\le x<2\\ 6-x,& x\ge 2\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}1+\tan x,& 0\le x<\pi /4\\ 3-\cot x,& \pi /4\le x<\pi \end{array}$

Define $h\left(x\right)$ by $h\left(x\right)=f\left(g\right(x\left)\right)$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(I)}& \text{If}h\left(x\right)\text{is continuous but not differentiable at}x=\frac{a\pi }{b},& \text{(P)}& 2\\ & \text{where}a,b\text{are co-prime, then}a+b\text{is}& & \\ \text{(II)}& \text{Number of points of non-differentiability of}|h\left(x\right)|\text{is}& \text{(Q)}& 5\\ \text{(III)}& \text{If range of}h\left(x\right)\text{is}(-\infty ,k],\text{then the value of}k\text{is}& \text{(R)}& 1\\ \text{(IV)}& \text{The value of}{h}^{\prime \prime }\left(x\right)\text{at}x=\pi /4^{-}\text{is}4m\text{. Then}m\text{is}& \text{(S)}& 4\end{array}$

Which of the following is CORRECT combination?

• A. $\left(I\right)\to \left(Q\right),\left(II\right)\to \left(P\right),\left(III\right)\to \left(R\right),\left(IV\right)\to \left(S\right)$
• B. $\left(I\right)\to \left(S\right),\left(II\right)\to \left(Q\right),\left(III\right)\to \left(R\right),\left(IV\right)\to \left(P\right)$
• C. $\left(I\right)\to \left(P\right),\left(II\right)\to \left(Q\right),\left(III\right)\to \left(R\right),\left(IV\right)\to \left(S\right)$
• D. $\left(I\right)\to \left(Q\right),\left(II\right)\to \left(P\right),\left(III\right)\to \left(S\right),\left(IV\right)\to \left(R\right)$

Answer 1

The correct option is D $\left(I\right)\to \left(Q\right),\left(II\right)\to \left(P\right),\left(III\right)\to \left(S\right),\left(IV\right)\to \left(R\right)$

$f\left(x\right)=\{\begin{array}{cc}x+2,& 0\le x<2\\ 6-x,& x\ge 2\end{array}$

and $g\left(x\right)=\{\begin{array}{cc}1+\tan x,& 0\le x<\pi /4\\ 3-\cot x,& \pi /4\le x<\pi \end{array}$


$f\left(g\right(x\left)\right)=\{\begin{array}{cc}3+\tan x,& 0\le x<\pi /4\\ 3+\cot x,& \pi /4\le x<\pi \end{array}$


$|h\left(x\right)|=|f\left(g\right(x\left)\right)|$


From above graph,
$h\left(x\right)$ is continuous but not differentiable at $x=\pi /4$
Also, $|h\left(x\right)|$ is not differentiable at two points.

Range of $h\left(x\right)$ is $(-\infty ,4]$

$h^{\prime \prime }\left(x\right)$ at $x=\pi /4$ is $2\times {\sec }^{2}x\times \tan x=4$