Find minimum of max function $f:\mathbb{R^+}\times\mathbb{R}\rightarrow\mathbb{R},\ f(x_1,x_2) = max\{1-\sqrt{x_1}, |x_2|\}$

Question

Is there any closed-form way of finding it using gradient and Hessian? I tried branching my function into

$2f(x_1,x_2) = \left\{ \begin{array}{ll} 1-\sqrt{x_1}\ - x_2\ + |1 - \sqrt{x_1} + x_2|, \ x_2 \le 0 \\ 1-\sqrt{x_1}\ + x_2\ + |1 - \sqrt{x_1} - x_2|, \ x_2 > 0 \end{array} \right.$

and find the subdifferential

$\frac{\partial f}{\partial x_1}(x_1,x_2)=0 \\ \frac{\partial f}{\partial x_2}(x_1,x_2)=\left\{ \begin{array}{ll} 0,\ x_2 \le 0 \\ 1,\ x_2 > 0 \end{array} \right.$

but plotting the function seems to show something completely different

Answer 1

$f(x_1,x_2) \geq |x_2| \geq 0$ and $f(1,0)=0$ so the minimum value is 0.