I wish to solve the following using Numerical Methods: $$ \bar{m} = \underset{m \geq 0}{\text{argmin}} \left( \int_a^b \left( \frac{1}{\left(\sum_{i=1}^M \left(c_i^\alpha \cdot n^2 y^{-m-1} \cdot \int_0^1 \left( 1-x\right)^{n-1} \cdot x^m \cdot \left(1 - z \cdot x \right)^{n-1} dx\right)\right)^{\frac{1}{2}}} - \frac{1}{\sqrt{f(x)}} \right)^2 dx \right)^{\frac{1}{2}} $$

I already know the following:

- $c_i^\alpha$ for all $i$
- M
- n
- y
- z
- f(x); note: f(x) is quite hideous, so I spared writing out the formula. it doesn't contain any of the parameters listed in (1 through 5). It's just ugly.
- and interval [a,b]

I will also point out that $\int_0^1 \left( 1-x\right)^{n-1} \cdot x^m \cdot \left(1 - z \cdot x \right)^{n-1} dx$ comes from the Euler type Gaussian hypergeometric function

What I have considered doing:

- Try using lagrange multiplier on this. The trouble is is that I am overwhelmed by the math and am not sure how to use the lagrange multiplier on this problem.

any help pointing me into the right direction into solving this problem (via lagrange multiplier or something else) would be great.

-Jas