The question is as in the title: what's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$? (Assume suitable smoothness conditions.)
A problem in the textbook for the course I am TEACHING (not taking) reduces to minimizing $w_1^2 + w_2^2 + w_3^2$ subject to $w_1 + w_2 + w_3 = 0, w_1 + 2w_2 + 3w_3 = 1$. Of course there's nothing special about the number $3$ here, and so one can ask for the minimum of $\sum_{i=1}^n w_i^2$ subject to $\sum_{i=1}^n w_i = 0, \sum_{i=1}^n iw_i = 1$. At least when $n = 3, 4, 5$, we get $w_i = c_n(i-(n+1)/2)$, for some constant $c_n$ which depends on $n$. So for fixed $n$, $w_i$ is a linear function of $i$. (This is a bit of an annoying computation, so I won't reproduce it here.)
So it seems like there should be a continuous analogue of this. If we have $ \int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1 $ and $f(x)$ is linear, then we get $f(x) = 12(x-1/2)$, and $\int_0^1 f(x)^2 \: dx = 12$. Is this the function satisfying these integral conditions with smallest $\int_0^1 f(x)^2 \: dx$? That is, is it the case that $ \int_0^1 f(x)^2 \: dx \ge 12 $ for every $f(x)$ satisfying the two conditions above and whatever smoothness conditions are necessary?
I've tagged this calculus-of-variations because that's what it looks like to me. But I don't know the calculus of variations, which is why I can't just solve the problem myself.