Question

If $y={\cos }^{-1}\cos (|x|-f(x)),$ where $f(x)=\{\begin{array}{ccc}1,& if& x>0\\ -1,& if& x<0\\ 0,& if& x=0\end{array}$ Then, ${(dy/dx)}_{x=\frac{5\pi }{4}}$ is equal to

• A. $-1$
• B. $1$
• C. $0$
• D. can't be determined

Answer 1

Answer: (B)

$\therefore$ $y={\cos }^{-1}\cos (x-1),x>0$
$\Rightarrow$ $y=x-1,0\le x-1\le \pi$
$\Rightarrow$ $y=x-1,1\le x\le \pi +1$
$\therefore$ $y=x-1,1\le x\le \pi +1$
At $x=\frac{5\pi }{4}\in [1,\pi +1]$
$\Rightarrow$ $\frac{dy}{dx}=1$
$\Rightarrow$ ${\left(\frac{dy}{dx}\right)}_{x=\frac{5\pi }{4}}=1$