Compute llg-fll^2 in C(1,3) Real Linear Space

[Cassi]

Homework Statement

In the real linear space C(1, 3) with inner product (f, g) = intergal (1 to 3) (f(x)g(x))dx, let f(x) = 1/x. Knowing that g = (1/2)log3 is the constant polynomial g that is nearest to f. Compute llg-fll² for this g.

Homework Equations

The Attempt at a Solution

I devised llg-fll = llgll - llfll = sqrt(g, g) - sqrt(f,f). Therefore, llg-fll² = (g, g) - (f, f)
Using the equations, this equals integral(1 to 3)(1/2log3)²dx - integral (1 to 3)(1/x)²dx
Simplifying I get (1/2log3)²x evaluated from 1 to 3 +1/x evaluated from 1 to 3 = log²3-2/3

My book says the answer is supposed to be 2/3 - 1/2log²3 but I do not get this. Where am I going wrong?

 

[pasmith]

Homework Statement

In the real linear space C(1, 3) with inner product (f, g) = intergal (1 to 3) (f(x)g(x))dx, let f(x) = 1/x. Knowing that g = (1/2)log3 is the constant polynomial g that is nearest to f. Compute llg-fll² for this g.

Homework Equations

The Attempt at a Solution

I devised llg-fll = llgll - llfll = sqrt(g, g) - sqrt(f,f).

This is wrong. $$\|g - f\|^2 = (g,g) - 2(f,g) + (f,f).$$ Hence $$\|g - f\| = \sqrt{(g,g) - 2(f,g) + (f,f)}.$$ However you should in finding $g$ have needed to calculate $$\|g - f\|^2 = \|f - g\|^2 = \int_1^3 (f(t) - g(t))^2\,dt$$ anyway, so you can simply substitute the found $g$.