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

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In summary, The conversation discusses the process of finding the coefficients a, b, and c that minimize the norm of the function f(x)=x^3-c-bx-ax^2 under L_2[-1,1]. One method proposed is equating f to the sum of <f,phi_k>phi_k where the phi_k are orthonormal functions. However, this approach leads to two different solutions for b, indicating a potential error. The other method suggested is using Legendre polynomials, as they are defined on the interval [-1,1]. A simpler approach is also mentioned, which involves integrating f^2 and solving for the coefficients that give a minimum. The result of this approach is a=c=0
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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 [tex]f=\sum_k <f,\phi_k>\phi_k[/tex] 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.
 
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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?
 
  • #3


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

What is the domain of the function?

The domain of the function f(x) = x^3-c-bx-ax^2 is all real numbers, since there are no restrictions on the values of x.

What is the range of the function?

The range of the function f(x) = x^3-c-bx-ax^2 depends on the values of a, b, and c. Without any restrictions on these values, the range can be any real number.

How do I find the x-intercepts of the function?

To find the x-intercepts of f(x) = x^3-c-bx-ax^2, set f(x) equal to 0 and solve for x using the quadratic formula. The resulting values of x will be the x-intercepts.

What are the possible types of stationary points for this function?

The possible types of stationary points for f(x) = x^3-c-bx-ax^2 are local minimums, local maximums, and points of inflection. The type of stationary point depends on the values of a, b, and c.

Can this function have multiple critical points?

Yes, this function can have multiple critical points. The number of critical points depends on the values of a, b, and c. For example, if a = 0, there will be only one critical point. But if a, b, and c are all non-zero, there can be multiple critical points.

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