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Part CCSIR NET June 2024absolute-value-multiplied-is-smooth-once

Absolute value multiplied is smooth once

Define f:R→Rf : \mathbb{R} \to \mathbb{R} by f(x) = x|x|. Which of the following statements are true?

  1. A.f is continuous on R\mathbb{R}✓ Correct
  2. B.f is differentiable on R\mathbb{R}✓ Correct
  3. C.f is differentiable only at 0
  4. D.f is not differentiable at 0

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: Differentiable ⇒ continuously differentiable

More on this topic

The chapter behind this: Differentiation — MVT, Darboux, Taylor and the classic spoilers — free to read

From Continuity and Differentiation › Differentiability, mean value theorems, Taylor, L'Hôpital

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