Let be a function defined by . The set of all points, where is not differentiable, is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: D
Solution
f(x) = max{x, x³}. The two functions x and x³ switch dominance at their intersection points: x = x³ gives x ∈ {-1, 0, 1}. Between these points: for x < -1, x³ < x so f = x; for -1 < x < 0, x³ > x so f = x³; for 0 < x < 1, x³ < x so f = x; for x > 1, x³ > x so f = x³. At each of x = -1, 0, 1, f switches between x and x³. At these points the left-hand and right-hand derivatives differ (e.g., at x = 0: left derivative from x³ is 3(0)² = 0 but right derivative from x is 1), making f non-differentiable at all three points {-1, 0, 1}.
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