If functions $f$ and $g$ are continuous on $[a,b]$ differentiable on $(a,b)$, and $f'(x) = g'(x)$ on $(a,b)$, then there exists a real number $K$ such that $f(x) = g(x) + K$ for all $x\in [a,b]$.
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Hint: Set $h=f-g$. Show that $h$ is continuous on $[a,b]$ and that $h'$ is identically zero on $(a,b)$. Then use the result of a previous question of yours.