Question

Given f '(x) = (2 - x)(6 - x), determine the intervals on which f(x) is increasing or decreasing. (2 points)

Answer 1

Answer:

Decreasing (2, 6); increasing on (-∞, 2) U (6, ∞)

Step-by-step explanation:

To determine where $f(x)$ is increasing or decreasing, we set $f'(x)=0$ and check for the intervals.

We see that $f'(x)=0$ when either $x=2$ or $x=6$. Therefore, we'll need to check the intervals $(-\infty,2)$, $(2,6)$, and $(6,\infty)$

For the interval $(\infty,2)$, we can pick $x=0$ . This means that$f'(0)=(2-0)(6-0)=(2)(6)=12>0$, showing that $f(x)$ increases on the interval $(-\infty,2)$

For the interval $(2,6)$, we can pick $x=4$. This means that $f'(4)=(2-4)(6-4)=(-2)(2)=-4$, showing that $f(x)$ decreases on the interval $(2,6)$

For the interval $(6,\infty)$, we can pick $x=7$. This means that $f'(7)=(2-7)(6-7)=(-5)(-1)=5>0$, showing that $f(x)$ increases on the interval $(6,\infty)$

Therefore, $f(x)$ is increasing on $(-\infty,2)\cup(6,\infty)$ and is decreasing on $(2,6)$.