I would like to numerically estimate (and plot) the continous functions $h(x)$ and $p(x)$ in the interval $[x_1;x_2]$, that minimize the following expression:
$$F=\int_{x_1}^{x_2}\frac{1}{1+\left(\frac{p(x)}{x}\right)^{h(x)} } dx$$
where $p(x_1) \leq p(x) \leq p(x_2)$, $h(x)>0$ and $p'(x)>0$. The values are: $$x_1=4.5$$ $$x_2=14$$ $$p(x_1)=3.5$$ $$p(x_2)=4.6$$ $$h(x_1)=2.8$$ $$h(x_2)=2.4$$
Is it possible to do it in Maple or Mathematica?
Kind regards Rasmus
I can make $F$ as small as I desire. Suppose that for $x>4.51$ and $x<13.99$, I made $p(x)=-1.00000000001\cdot x$ and $h(x)=1$. Do you see the problem?
Edit
Even after your new conditions, your problem is still broken. Minimizing $F$ requires us to now maximize $p(x)$ and $h(x)$. As $p(x),h(x)\to\infty$, $F\to0$, which is the lower bound for the integral. As such, this question is pointless, because I can always make $p(x)$ and $h(x)$ larger to make $F$ smaller.