Let f : R2 → R be a twice-differentiable function with non-zero second partial derivatives. Suppose that for every x ∈ R, there is a unique value of y, say y∗(x), that solves the problem
max y∈R f(x, y). Then y * is increasing in x if
If y*(x) increases with x, this means that the effect of cross partial derivatives exceeds the effect of own partial derivatives, thus c. For both options a and b this is not the case, the effect own partial derivatives exceed the effect of cross partial derivatives.