Suppose $f : [ 0,1 ] \rightarrow \mathbb { R }$ is differentiable with $f ( 0 ) = 0$. If $\left| f ^ { \prime } ( x ) \right| \leq f ( x )$ for all $x \in [ 0,1 ]$, then show that $f ( x ) = 0$ for all $x$.
Suppose $f : [ 0,1 ] \rightarrow \mathbb { R }$ is differentiable with $f ( 0 ) = 0$. If $\left| f ^ { \prime } ( x ) \right| \leq f ( x )$ for all $x \in [ 0,1 ]$, then show that $f ( x ) = 0$ for all $x$.