Mathematics

Mathematics

Problem

For m>0m>0, consider the family of functions gm(x)=∣x∣mg_m(x)=|x|^{m}. At x=0x=0, the derivative gm′(0)g_m'(0) exists under which of the following conditions on mm? a) It exists only when m≥1m\ge 1. b) It exists only when m>1m>1. c) It exists only when m=1m=1. d) It exists only when 0<m<10<m<1.
Intuitive explanation

The key insight is that differentiability at x=0x=0 requires the left and right derivatives to exist and match. Since gm(x)=∣x∣mg_m(x)=|x|^m is symmetric around zero, you need to check whether the limit lim⁡h→0∣h∣mh\lim_{h \to 0} \frac{|h|^m}{h} behaves the same from both sides. This boils down to asking: for which values of mm does the expression ∣h∣mh\frac{|h|^m}{h} approach a single finite value as h→0+h \to 0^+ and h→0−h \to 0^-?

Intuitive explanation

The key insight is that differentiability at x=0x=0 requires the left and right derivatives to exist and match. Since gm(x)=∣x∣mg_m(x)=|x|^m is symmetric around zero, you need to check whether the limit lim⁡h→0∣h∣mh\lim_{h \to 0} \frac{|h|^m}{h} behaves the same from both sides. This boils down to asking: for which values of mm does the expression ∣h∣mh\frac{|h|^m}{h} approach a single finite value as h→0+h \to 0^+ and h→0−h \to 0^-?

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