The function f(x)=| | is: 1. Everywhere continuous and differenti… - Vidyadip

Question

The function $\text{f(x)}=|\cos\text{x}|$ is:

Everywhere continuous and differentiable.
Everywhere continuous but not differentiable at $(2\text{n}+1)\frac{\pi}{2},\text{n}\in\text{Z}$
Neither continuous nor differentiable at $(2\text{n}+1)\frac{\pi}{2},\text{n}\in\text{Z}$
None of these.

✓

Answer

Everywhere continuous but not differentiable at $(2\text{n}+1)\frac{\pi}{2},\text{n}\in\text{Z}$

Solution:
As cos x is even function it is continuous everywhere but not differentiable at $(2\text{n}+1)\frac{\pi}{2},\text{n}\in\text{Z}$
$\cos\Big[(2\text{n}+1)\frac{\pi}{2}=\cos\Big(\text{n}\pi+\frac{\pi}{2}\Big)=-\sin\text{n}\pi$
For n as an integer $\Rightarrow\sin\text{n}\pi=0$
For n as rational $\Rightarrow\sin\text{n}\pi=-1$

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