Question
The number of values of $$x\, \in \left[ {0,\,2} \right]$$ at which the real function $$f\left( x \right) = \left| {x - \frac{1}{2}} \right| + \left| {x - 1} \right| + \tan \,x$$ is not finitely differentiable is :
A.
2
B.
3
C.
1
D.
0
Answer :
3
Solution :
The doubtful points are $$x = \frac{1}{2},\,1,\,\frac{\pi }{2}$$
$$\left| {x - 1} \right|$$ is differentiable at $$\frac{1}{2},\,\frac{\pi }{2}$$ but not differentiable at $$1.$$
$$\left| {x - \frac{1}{2}} \right|$$ is differentiable at $$1,\,\frac{\pi }{2}$$ but not differentiable at $$\frac{1}{2}$$
Remember $$\left| {x - a} \right|$$ is continuous everywhere, and differentiable everywhere except at $$x=a$$
$$\tan \,x$$ is differentiable at $$\frac{1}{2},\,1$$ but not differentiable at $$\frac{\pi }{2}$$
$$\therefore \,f\left( x \right)$$ is not differentiable at $$\frac{1}{2},\,1,\,\frac{\pi }{2}$$