Let $0<x\leq \frac{1}{2}$ define the function : $$f(x)=x^{\sqrt{2(1-x)}}+(1-x)^{\sqrt{2x}}$$
And let $f(x_0)$ be the minimum of the function on $(0,1/2)$
Then we have : $$f(x)+f(x_0)\leq 2$$
The maximum of the function is around $1.000150515\cdots$ and the minimum is around $0.9989495662\cdots$
I have tried derivative and Newton's method but it is not elegant so if you have a trick for this problem...
Thanks a lot .