CPM Homework Help : APCALC Problem 5-169

The formula $h\left(x\right) = \left(x – 2\right)\left(ax – 1\right)^{2}$ defines a family of functions, each corresponding to a different value of the parameter, $a$ . Determine the values of $x$ for which this family of functions has a relative maximum or minimum. Your answers will be in terms of $a$ .

Step 1:

Notice that $h\left(x\right)$ is a cubic function. Visualize its graph.

$\text{There will be two roots: }x=2\text{ and }x=\frac{1}{a}.$

$\text{Notice that }x=\frac{1}{a}\text{ is the location of a DOUBLE ROOT;}$

consequently, it is the location of a local max or min of $h\left(x\right)$ .

Step 2:

Since $h\left(x\right)$ is a cubic function, there should be another local max or min. Find its location by setting $h '\left(x\right) = 0$ and solving for $x$ .

Step 3:

Recall that maxima and minima are $y$ -values. So evaluate $h\left(x\right)$ to find the corresponding $y$ -values.