
Problem 11.1. For any real number $b$ denote by $f(b)$ the maximal value of $\left|\sin x+\frac{2}{3+\sin x}+b\right|$. Find the minimal value of $f(b)$.


Solution: Substitute $t=\sin x$ and $g(t)=t+\frac{2}{3+t}+b$. Since $g(t)$ is an increasing function in the interval $[-1,1]$, it follows that $f(b)=\max (|g(-1)|,|g(1)|)=\max \left(|b|,\left|b+\frac{3}{2}\right|\right)$. Now from the graph of the function $f(b)$ we conclude that $\min f(b)=f\left(-\frac{3}{4}\right)=\frac{3}{4}$.
