1. Optmization Problem

Determine the value of the constant k which best approximates a specified function f(x) on the interval 𝑎≤𝑥≤𝑏. Use as a cost function to be minimized:

Verify that a minimum is achieved by checking the 2nd order optimality condition.

I rewrote L(k) and then tried to find the value of k such that dL(k)/dk = 0. Am I allowed to simplify like this?

2. Re: Optmization Problem

Hey Sylvan.

You can write it like this but you should be careful in your algebra.

The integral you evaluate as coefficients of your k^2 and k terms will not be the ones you got in your derivative term.

You will need to evaluate these integrals and if the x and k variables are independent then they will become the proper coefficients just like d/dx ax^2 = 2ax and not 2x.

3. Re: Optmization Problem

k is a constant, not a variable, which is why I pulled it out of the integral.

I tried evaluating the integrals again:

And I apologize for the double thread, this one didn't show up. I will try to delete the duplicate.

4. Re: Optmization Problem

If you are asking "what value of k ..." then you must treat k as a variable! If it were a constant then it could have only one value.

5. Re: Optmization Problem

Minor typo, since $x^b_a=b-a$, the answer should be $k=\frac{f(b)-f(a)}{b-a}$.