### Home > APCALC > Chapter 5 > Lesson 5.5.2 > Problem5-169

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 $a$.

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 $h\left(x\right)$.

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 $x$.

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