MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
The function f(x)=∣x∣ is not differentiable at x=0. Which statement best explains why the derivative fails to exist at that point?
(a) Because ∣x∣ is defined by two different formulas on either side of x=0, its derivative cannot exist at the point where the pieces meet.
(b) The slopes of the secant lines approaching x=0 from the left and right are unequal, so the two-sided limit defining f′(0) does not exist.
(c) The graph of f has a vertical tangent at x=0, producing an infinite derivative.
(d) The function is not continuous at x=0, preventing the derivative from being defined there.