AHSME 1980 Problem 22

For each real number $x$, let $f(x)$ be the minimum of the numbers $4x+1, x+2$, and $-2x+4$. Then the maximum value of $f(x)$ is $\text{(A)} \ \frac{1}{3} \text{(B)} \ \frac{1}{2} \text{(C)} \ \frac{2}{3} \text{(D)} \ \frac{5}{2} \text{(E)}\ \frac{8}{3}$