Problem 2

On $[-1,1]$, $f$ is continuous. Both $\sqrt{3-x}$ and $-\sqrt{x+1}$ decrease as $x$ increases, so $f$ decreases. Its endpoint values are $f(-1)=\sqrt2$ and $f(1)=0$; therefore there is exactly one point where $f(x)=1/2$, and the strict inequality holds precisely before that point.