If $f$ and $g$ are continuous, then max(f, g) is continuous and differentiable If $f$ and $g$ are continuous on $[a, b]$ and differentiable on $(a, b)$, then $\max(f, g)$ is continuous on $[a, b]$ and differentiable on $(a, b)$.

I'm asked to either prove or disprove this statement.  So far I'm thinking that it's true because if a function is continuous and differentiable on an interval, it takes on it's max/min over the interval, but I don't know if this is correct, and if it is, I don't know how to prove it.  Any help would be appreciated.
Idea

Consider $\lvert x \rvert = \max\{x, -x\}$.
Can use the fact that max (f,g) = {f + g + |f-g|}/2