Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ be a continuous function which is differentiable on $( 0,1 )$. Prove that either $f$ is a linear function $f ( x ) = a x + b$ or there exists $t \in ( 0,1 )$ such that $| f ( 1 ) - f ( 0 ) | < \left| f ^ { \prime } ( t ) \right|$.