Find stationary point of non-canonical form in calculus of variation

Question

Let $x \in [0, 1]^{10}$ be a ten dimensional vector. I want to maximize the functional with the form: $$G(f) = \int_{0 \leq y_1 \leq x_1 \leq 1, x_i, y_i \in [0, 1], i \neq 1} f(x) f(y) (1 + 3 x_1^2 - x_1^3 - 6 x_1 y_1+ 3 x_1^2 y_1 + 3 y_1^2 - 3 x_1 y_1^2 + y_1^3)dx dy$$ with constraints $$\int_{x} f(x) dx = 0 \text{ and } \int_{x} f^2(x) dx = 1,$$ where $x_1, x_2$ are the first and second element of $x$. This isn't canonical form of euler lagrangian. I guess the maximizer should be $f(x) = \sqrt{12}(x_1 - 1/2)$. But i don't know how to differentiate $G$. Any hints or reference for this problem? thanks.