1980 AHSME Problems/Problem 22

Problem

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

$\text{(A)} \ \frac{1}{3} \qquad  \text{(B)} \ \frac{1}{2} \qquad  \text{(C)} \ \frac{2}{3} \qquad  \text{(D)} \ \frac{5}{2} \qquad  \text{(E)}\ \frac{8}{3}$

Solution

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 \[f(x)=\left\{ \begin{matrix} 4x+1 & x<\frac{1}{3} \\                                            x+2   & \frac{1}{3}>x>\frac{2}{3} \\                                           -2x+4 & x>\frac{2}{3} \end{matrix}\right.\] Which attains a maximum at $\boxed{(E)\ \frac{8}{3}}$

See also

1980 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 23
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