Read the two statements and choose the correct option. I. f(x) = x^2 sin 1/x for x ≠ 0, with f(0) = 0, is differentiable at x = 0. II. g(x) = x sin 1/x for x ≠ 0, with g(0) = 0, is differentiable at x = 0.

Question 121

HardOriginal question

Read the two statements and choose the correct option.
I. $f(x) = x^{2} \sin \frac{1}{x}$ for $x \ne 0$ , with $f(0) = 0$ , is differentiable at $x = 0$ .
II. $g(x) = x \sin \frac{1}{x}$ for $x \ne 0$ , with $g(0) = 0$ , is differentiable at $x = 0$ .

Only ICorrect answerYour answerOnly IICorrect answerYour answerBoth I and IICorrect answerYour answerNeither I nor IICorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

I: $\frac{f(h) - f(0)}{h} = h \sin \frac{1}{h} \to 0$ , so $f'(0) = 0$ . True.
II: $\frac{g(h) - g(0)}{h} = \sin \frac{1}{h}$ , which keeps swinging between $-1$ and 1 and has no limit. False, though $g$ is continuous at 0.

Worth noting

Continuity at 0 does not give differentiability: the difference quotient of $g$ never settles.

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