lagrange multipliers to find minimizing pdf

Question

Chapter 14, Question 7 of Van Der Vaart's Asymptotic Statistics is to find the probability density $f$ minimizing $\int y^2f(y)dy$ over all densities $f$ bounded by $1$. Question 8 is to find the pdf minimizing $\int f^2(y)dy$ over all $f$ with mean $0$ and variance $1$. I have an idea how to solve these problems with a lot of calculus but my question is how to solve them using the hint he suggests. He says to use the following fact. "Suppose $\phi:\mathcal{F}\mapsto\mathbb{R}$ and $\psi:\mathcal{F}\mapsto\mathbb{R}^m$ are arbitrary maps on an arbitrary set $\mathcal{F}$ and we wish to find the minimum value of $\phi$ over $\{f\in\mathcal{F}:\psi(f)=0\}$. If the map $f\mapsto\phi(f)+a^T\psi(f)$ attains its min over $\mathcal{F}$ at $f_a$ for each fixed $a$ in an arbitrary set $A$, and there exists $a_0\in A$ such that $\psi(f_{a_0})=0$, then the desired min is $\phi(f_{a_0})$." I can see that the constraints like mean $0$ and variance $1$ correspond to $\psi$, and $\phi$ is the objective like $\int f^2(y)dy$, but I don't see how the hint helps. Finding the minimizing $f_a$ looks as hard as the original problem.