For a convex function $F$ of one variable you can always find the minimum by simple bisection: if you know that the minimum is attained on $[a,b]$ (in your case $a=0$ and the initial $b$ can be found going to the right geometrically until the current value is greater than the previous one), then you can split $[a,b]$ into 4 equal parts by the points $a<x<y<z<b$ and look at the values at those points to choose a smaller interval.
If $F(a)<F(x)$, switch to $[a,x]$, else
if $F(x)<F(y)$, switch to $[a,y]$, else
if $F(y)<F(z)$, switch to $[x,z]$, else
if $F(z)<F(b)$, switch to $[y,b]$, else
switch to $[z,b]$.
This approach is somewhat crude but quick and once you are close enough to the minimum, you can find it exactly because you will know which brackets are involved.