MathDB
differentiability of functions at 0

Source: LIMIT 2019 CCS2 P1

April 28, 2021
function

Problem Statement

Which of the following functions are differentiable at x=0x=0? <spanclass=′latex−bold′>(A)</span> f(x)={tan⁡−1(1∣x∣)if x≠0π2if x=0<span class='latex-bold'>(A)</span>~f(x)=\begin{cases}\tan^{-1}\left(\frac1{|x|}\right)&\text{if }x\ne0\\\frac\pi2&\text{if }x=0\end{cases} <spanclass=′latex−bold′>(B)</span> f(x)=∣x∣1/2x<span class='latex-bold'>(B)</span>~f(x)=|x|^{1/2}x <spanclass=′latex−bold′>(C)</span> f(x)={x2∣cos⁡πx∣if x≠00if x=0<span class='latex-bold'>(C)</span>~f(x)=\begin{cases}x^2\left|\cos\frac{\pi}x\right|&\text{if }x\ne0\\0&\text{if }x=0\end{cases} <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
Back to ProblemsView on AoPS