Can we find a function g(x) such that it minimizes $\int_{5}^{10} \frac {g'}{\sqrt{g}} \,dx$, subject to the boundary conditions g(5) = 3, g(10)= 8, as well as the constraint g(7) = 6.
I this question solvable? I thought of using lagrangian multipliers but I can't think of how to do so with the constraints and function we want to minimize having different input spaces.
$\begin{align} \int_5^{10}\frac{g'(x)}{\sqrt{g(x)}} \,dx&=\bigg[2\sqrt{g(x)}\bigg]_5^{10}\!\!=2\bigg(\!\sqrt{g(10)}-\sqrt{g(5)}\bigg)=\\ &=2\left(\sqrt8-\sqrt3\right)=2\left(2\sqrt2-\sqrt3\right)\,. \end{align}$
Every positive differentiable function $\;g(x)\;$ on the interval $\;\left[5,10\right]\;$ such that $\;g(5)=3\;$ and $\;g(10)=8\;,\,$ gives the same value of the integral, so it does not make sense to find the one that minimizes it.