Answer
True
Work Step by Step
An inflection point occurs where the concavity changes sign.
If $f'$ is increasing on $[0,1]$, it means $f''(x)\geq 0$ on $[0,1]$ and if $f'$ is decreasing on $[1,2]$, it means $f''(x)\leq 0$ on $[1,2]$. So the concavity changes sign at $x=1$, which means $x=1$ is an inflection point.
The statement is true.
