How is f(x) =|x-1| differentiable at x=2?

Give details Explaination.

Expert Solution

f(x) is continuous everywhere. To be differentiable at x=x1, the derivative as x→x1 from the left has to equal the derivative as x→x1 from the right.

For f(x) = |x-1|, f(x) = x-1 for x>=1 and 1-x for x<1. Since 2>=1, f’(2) = 1 whether you approach 2 from the left or the right. The same reasoning holds for every value of x except for 1.

f’(1) = 1 from the right and f’(1) = -1 from the left, so f(x) is not differentiable at x=1.

Arjoma Karan answered 2 years ago

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