Answer
False
Work Step by Step
Consider $f(x)=6,g(x)=x, a=-4, b=4$
Therefore,
$f(x)\geq g(x)$ on $(-4,4)$
However $f'(x) \leq g'(x)$, since $f'(x)=0$ and $g'(x)=1$
Hence, the statement is false.
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