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

Absolute value multiplied is smooth once

Define by f(x) = x|x|. Which of the following statements are true?

  1. A.f is continuous on ✓
  2. B.f is differentiable on ✓
  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.

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50 are analysed free — try those first.

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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