Which of these functions are differentiable at x = 0? I. f(x) = | x | sin x II. f(x) = | sin x | III. f(x) = cos | x | IV. f(x) = x + | x |

Question 106

MediumOriginal question

Which of these functions are differentiable at $x = 0$ ?
I. $f(x) = \lvert x \rvert \sin x$
II. $f(x) = \lvert \sin x \rvert$
III. $f(x) = \cos \lvert x \rvert$
IV. $f(x) = x + \lvert x \rvert$

III onlyCorrect answerYour answerI, II and III onlyCorrect answerYour answerI and III onlyCorrect answerYour answerI, III and IV onlyCorrect answerYour answer

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CorrectNot quiteAnswer C

Correct answer: C

I is $-x \sin x$ on the left and $x \sin x$ on the right, both with slope 0 at 0. III is $\cos x$ , since cosine is even. Both are differentiable.
II has slopes $-\cos 0 = -1$ and $\cos 0 = 1$ , a corner. IV is 0 on the left and $2x$ on the right, with slopes 0 and 2.

Worth noting

A factor that is 0 at the corner, like $\sin x$ in I, or an even outer function, like $\cos$ in III, smooths $\lvert x \rvert$ out.

Concept: Recognising differentiable functions
Typical timeTypical: 120 s
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The rule this testsDifferentiability

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