How to Minimize the Norm of f(x)=x^3-c-bx-ax^2 Under L_2[-1,1]?

[MathematicalPhysicist]

Assume I have the next function f(x)=x^3-c-bx-ax^2 and I am asked to find the coefficients a,b,c which minimizes the norm of f under L_2[-1,1].

All I need to do here is equate f=\sum_k <f,\phi_k>\phi_k where the phis are orthonormal functions, in this case simply 1,x,x^2,x^3, I am not sure this correct cause I found the next coefficients:
<f,1>=sqrt(-2a-2c/3)
<f,x>=sqrt(2/5-2/3 b)
<f,x^2>=sqrt(-2a/3-2c/5)
<f,x^3>=sqrt(2/7-2b/5)

But when equation I find two different solutions to b, so I suspect this is the wrong to solve this problem, any hints as to how to minimize this functional.

 

[MathematicalPhysicist]

OK I think I know why I didn't get it right, I should be using Legendre polynomial cause they are defined on this interval [-1,1].

Have I got it right this time?

 

[mathman]

Your approach seems unduly complicated. Why not simply integrate f² and find the values of a, b, c which gives a minimum? I tried it myself (no guarantee - I am lousy in arithmetic) and got a=c=0 and b=3/5.