Take two real differentiable convex functions, $f_1$ and $f_2$, defined on the unit interval $[0, 1]$. I want to find the global optimum of
$\min_{x \in [0,1]} a f_1(x) + b f_2(x)$
for given $a, b \in \mathbb{R}$. Is there a simple solution to this?