Find the constant polynomial g closest to f

[Cassi]

Homework Statement

In the real linear space C(1, 3) with inner product (f,g) = integral (1 to 3) f(x)g(x)dx, let f(x) = 1/x and show that the constant polynomial g nearest to f is g = (1/2)log3.

Homework Equations

The Attempt at a Solution

I seem to be able to get g = log 3 but I do not know where the 1/2 comes from. Here is what I did:
Let f_n = summation (k=0 to n) (f, g_k) g_k
Therefore, (f, g_k) = integral (1 to 3) (1/t)(g_k)dt and (f, g₀) = 0
Hence, (f, g₁) = integral (1 to 3) (1/t)dt = log(t) evaluated from 1 to 3 = log (3) - log(1) = log(3) - 0 = log(3).
I don't see where the 1/2 comes from or where my mistake is?

 

[pasmith]

You need to find the constant $k$ which minimizes the distance between $f = 1/t$ and $g = k$. The distance is defined in terms of the inner product by $\| f - g \| = \sqrt{(f-g,f-g)}$. Since squaring is strictly increasing on the positive reals it suffices instead to minimize
$$\| f - g \|^2 = \int_1^3 (f(t) - g(t))^2\,dt.$$