Determine the minimum and maximum values of an integral subject to end conditions 
Determine the minimum and maximum values of the integral
$$\int_0^1 yy'dx$$
subject to the conditions $y(0)=0$ and $y(1)=1$.

There is no explicit dependence on $y$, so our Euler-Lagrange Equation is
$$F-\frac{d}{dx}\frac{\partial F}{\partial y'}=\text{constant}$$
However, when I do this, I get $y'y-y'y=0$. This tells me that
$$F=\frac{d}{dx}\frac{\partial F}{\partial y'}$$
Thus, the max/min correspond to those x-values where
$$\frac{dF}{dx}=\frac{d^2}{dx^2}\frac{\partial F}{\partial y'}=0$$
How do I take it from here? Did I do something wrong?

Hint: $yy'=\frac12(y^2)'$. ${}{}{}{}$
