Answer
False
Work Step by Step
\[\begin{align}
& \text{The statement is false, because the derivative }f'\left( 1 \right)\text{ can be 0} \\
& \text{on the interval }\left[ 0,2 \right],\text{ for example consider }\\
&f\left( x \right)=\text{ 2}{{\left( x-1 \right)}^{5}} \\
& f'\left( x \right)=10{{\left( x-1 \right)}^{4}} \\
& \text{Critical point }x=1 \\
& f'\left( 0 \right)=+,\text{ }f'\left( 2 \right)=+.\text{ The function }f\left( x \right)\text{ is increasing on} \\
& \left[ 0,2 \right]\text{ but }f'\left( 1 \right)=10{{\left( 1-1 \right)}^{4}}=0 \\
& \text{False} \\
\end{align}\]