Does this situation indicate that $g(x)>f(x)$?

Question

I have two functions $f(x)$ and $g(x)$ for $0<x<5$. By calculating the first and second derivatives I see that both functions are increasing and concave (not strictly concave), i.e. $f''<0$ and $g''<0$. Then If I have $$f(0)=g(0)=0$$ and $$f(5)=10\quad ,\quad g(5)=11.$$ Then, does this prove that $g$ is always grater than $f$?

Answer 1

No. Take $f(x)=10\sqrt[3]{x/5}$ and $g(x)=11\sqrt{x/5}$. You get something like this

You can easily come up with the same type of plot if $f''$ and $g''$ are positive

Answer 2

No. Take, for example, $$ f(x) = 10 - \frac{2}{5}(x-5)^2, \qquad g(x) = \frac{11}{5}\, x. $$ (If you like, you can slightly modify $g$ to be strictly concave.)