KCET · Maths · Continuity and Differentiability

The function \(f(x)=|\cos x|\) is

A Everywhere continuous and differentiable
B Everywhere continuous but not differentiable at odd multiples of \(\pi / 2\)
C Neither continuous nor differentiable at \((2 n+1) \frac{\pi}{2}, n \in Z\)
D Not differentiable everywhere

Verified Solution

Answer & Solution

Correct Answer

(B) Everywhere continuous but not differentiable at odd multiples of \(\pi / 2\)

Step-by-step Solution

Detailed explanation

\(f(x)=|\cos x|\)
Graph of \(f(x)=|\cos x|\)

From graph we can see that \(f(x)\) is not differentiable at points \(\frac{\pi}{2}, \frac{3 \pi}{2}, \frac{-\pi}{2} \ldots\). . Also, it is everywhere continuous.
i.e. At \(x=(2 n+1) \frac{\pi}{2}, n \in Z\)
\(f(x)\) is not differentiable but, continuous everywhere.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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