jmc

We plot the lines $y=2x+2,$ $y=\frac{1}{2}x+1,$ and $y=-\frac{3}{4}x+7.$



The intersection of lines $y=2x+2$ and $y=\frac{1}{2}x+1$ is $\left(-\frac{2}{3},\frac{2}{3}\right),$ and the intersection of lines $y=\frac{1}{2}x+1$ and $y=-\frac{3}{4}x+7$ is $\left(\frac{24}{5},\frac{17}{5}\right).$

We can show that $f(x)$ is increasing on the interval $\left(-\infty ,\frac{24}{5}\right],$ and decreasing on the interval $\left[\frac{24}{5},\infty \right).$ Thus, the maximum value of $f(x)$ is $f\left(\frac{24}{5}\right)=\boxed{\frac{17}{5}}.$