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ISI 2024 P4

Source: college entrance

May 12, 2024
ISI 2024functioncalculuslimits

Problem Statement

Let f:R→Rf: \mathbb R \to \mathbb R be a function which is differentiable at 00. Define another function g:R→Rg: \mathbb R \to \mathbb R as follows: g(x)={f(x)sin⁡(1x) if x≠0</br>0if x=0.g(x) = \begin{cases} f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\</br>0 &\text{if} ~ x = 0. \end{cases} Suppose that gg is also differentiable at 00. Prove that g′(0)=f′(0)=f(0)=g(0)=0.g'(0) = f'(0) = f(0) = g(0) = 0.
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