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**Endowed** True or False: If $\displaystyle f$ and $\displaystyle g $ are differentiable on $\displaystyle [a,b]$ and $\displaystyle |f'(x)| \le 1 \le |g'(x)|$ $\displaystyle \forall x \in (a,b)$, then $\displaystyle |f(x) - f(a)| \le |g(x) - g(a)|$ $\displaystyle \forall$ $\displaystyle x \in [a,b]$

Since $\displaystyle f,g$ are differentiable on $\displaystyle [a,b]$ they are also continuous on $\displaystyle [a,b]$....All i can derive from this problem so far. Help a brother continue. Thank you.