Let f: R → R be differentiable at c ∈ R and f(c) = 0. If g(x) = |f(x)|, then at x = c, g is:

Correct option is A. Differentiable iff f'(c) = 0

Let f: R → R be differentiable at c ∈ R and f(c) = 0. We have g(x) = |f(x)|. We want to determine the differentiability of g at x = c.

If f'(c) = 0, then f(x) ≈ 0 near x = c. Then |f(x)| ≈ 0 near x = c, which is differentiable at x = c.

If f'(c) ≠ 0, then f(x) changes sign near x = c. Then g(x) = |f(x)| will have a sharp corner at x = c, making it non-differentiable.

Therefore, g(x) is differentiable at x = c if and only if f'(c) = 0.