Question

The number of points at which the function $f\left(x\right)=|x-1|+|\cos x|+\tan (x+\frac{\pi }{4})$ does not have a derivative in the interval $(-1,2)$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$f\left(x\right)=\mid x-1\mid +\mid \cos x\mid +\tan (x+\frac{\pi }{4})$
Let $f_1\left(x\right)=\mid x-1\mid$
$f_1\left(x\right)=(x-1)$ if $x⩾1$
$f_1\left(x\right)=-x+1$ if $x<1$
$f_1^{\prime }\left(x\right)=1$ for $x>1$
$f_1^{\prime }\left(x\right)=-1$ for $x<1$
Hence $f_1\left(x\right)$ is not differentiable at $x=1$.
Now suppose $f_2\left(x\right)=\mid cosx\mid$
This is not differentiable at $x=n\pi +\frac{\pi }{2}$
And $f_3\left(x\right)=tan(x+\frac{\pi }{4})$,
which is not differentiable at $x=n\pi +\frac{\pi }{4}$
So, in the interval $(-1,2)$, $f\left(x\right)=f_1\left(x\right)+f_2\left(x\right)+f_3\left(x\right)$ is not differentiable at 3 points: $1,\frac{\pi }{2},\frac{\pi }{4}$