Let's $f(x)$ and $g(x)$ be continuous functions on $[0,1]$.
Show an example of the maximum of $f(x)$ and $g(x)$ does not equal to the maximum of $(f+g)(x)$ on $[0,1]$.
Now I have tried to find an example, but for each time, the maximum of $f(x) +$ the maximum of $g(x)$ would equal to the maximum of $(f+g)(x)$. For example, I would have $f(x) = x^2$ and $g(x) = x+2$. The maximum of $f(x)$ would be $1$ and the maximum of $g(x) = 3$, so $1+3 = 4$. However, when I have $(f+g)(x) = x^2 + x + 2$, the maximum of $(f+g)(x) = 4$. This would happen each time I used a continuous function for both $f(x)$ and $g(x)$. Any help would be greatly appreciated.