The problem as attached in the picture.
The Attempt at a Solution
So here's the problem that I was facing:
G(t) has stationary points at t = 0, t = -1/2, t = -4
They correspond to (0,0), (a/4, -a) and (16a,-8a) in the function G(t).
However, only (0,0) is a suitable stationary point in F(x,y).
Here are some of my conclusions:
Before the "condition" x = at2, y = 2at was applied, (x,y) was free to have any possible value.
However, with the restriction implemented, again let's imagine F(x,y) as a terrain with many mountains, pits and saddles. By implementing this restriction we only see what lies along the "track" of h(x,y), taking the highest mountain and the lowest pit as the maximum and minimum. But this is not the case, as in the whole map of F(x,y) there may very well be even higher mountains and even deeper pits that just lie outside the track..