Answer
False
Work Step by Step
Slope of line joining $(1,f(1))$ and $(3,f(3))$ is
$\frac{f(3)-f(1)}{3-1}=\frac{0-(-2)}{2}=1$
According to mean value theorem, there is at least one value of $cā(1,3)$ for which $f'(c)=1$
Hence, the given statement is false.