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**1. The problem statement, all variables and given/known data**

Given a Hilbert space $$V = \left\{ f\in L_2[0,1] | \int_0^1 f(x)\, dx = 0\right\},B(f,g) = \langle f,g\rangle,l(f) = \int_0^1 x f(x) \, dx$$ find the minimum of $$B(u,u)+2l(u)$$.

**2. Relevant equations**

In my text I found a variational theorem stating this minimization problem is equivalent to solving

$$

B(f,g)+l(f) = 0

$$

for fixed ##g## and all ##f##.

**3. The attempt at a solution**

Plugging in values yields

$$

\int_0^1f(g+x)\, dx = 0.

$$

But from here I'm not sure how to handle the answer. Any suggestions? I'm guessing either ##g=-x## or else ##f## must be orthogonal to the function ##g+x## on the interval ##[0,1]##. Unsure how to proceed.