smc

The first two given functions intersect at $\left(\frac{1}{3},\frac{7}{3}\right)$, and last two at $\left(\frac{2}{3},\frac{8}{3}\right)$. Therefore $$f(x)=\{\begin{array}{cc}4x+1& x<\frac{1}{3}\\ x+2& \frac{1}{3}>x>\frac{2}{3}\\ -2x+4& x>\frac{2}{3}\end{array}$$ Which attains a maximum at $\boxed{\frac{8}{3}}$