The minimum of $ f(x) = -(x+1/2 \sqrt{1-x})$ for $ x$ between $ 0$ and $ 1$ occurs at $ x=15/16$ with $ f(15/16)=-17/16$ . This is a function that Mathematica evaluates quickly, and I have plotted the function and its minimum values below:

Mathematica quickly evaluates

`Minimize[{-x - 1/2 Sqrt[1 - x] , x > 0, x < 1}, x] `

but is extremely slow in evaluating

`NMinimize[{-x - 1/2 Sqrt[1 - x] , x > 0, x < 1}, x] `

(In fact, Mathematica appears to hang when I try to evaluate the above.)

Why is this and how can I speed up the evaluation? Is it the way I specified the bounds? I have in mind somewhat more complicated examples but I’m hoping to understand this simple example first. I’m running 12.0 Student Edition on Windows 10.