Chapter 4.1, Problem 100E

SAT Scores by Income The following graph shown U.S. math SAT scores as a function of parents’ income level.

Income ($1,000)

The regression curve shown in given by

$f\left(x\right)=-0.0034x^2+1.2x+444\left(10\le x\le 180\right),$ where $f\left(x\right)$ is the average math SAT score of a student whose parents earn x thousand dollars per year.

a. Find $f\text{'}\left(x\right)$, and use your formula to compute $f\text{'}\left(100\right)$, stating its units of measurement. What does the answer say about math SAT scores?

b. Is $f\text{'}\left(x\right)$ increasingor decreasing with increasing x? What does your answer say about math SAT scores? [HINT: See Example 4.]