To analyze the behavior of the function [tex]\( g(x) = (x + 4)^4 (x - 9) \)[/tex] at the points [tex]\( x = -4 \)[/tex] and [tex]\( x = 9 \)[/tex], we need to determine whether the graph touches or crosses the [tex]\( x \)[/tex]-axis at these points. This can be done by examining the first derivative of the function, which provides information about the slope of the graph at these points.
We start by considering the points of interest:
1. Behavior at [tex]\( x = -4 \)[/tex]:
To determine what happens at [tex]\( x = -4 \)[/tex], calculate the first derivative [tex]\( g'(x) \)[/tex] and substitute [tex]\( x = -4 \)[/tex].
- We know that [tex]\( g(-4) = 0 \)[/tex], so [tex]\((-4 + 4)^4 (-4 - 9) = 0\)[/tex]. This confirms that [tex]\( -4 \)[/tex] is a root of the function.
- Next, check the value of the first derivative [tex]\( g'(-4) \)[/tex]. If [tex]\( g'(-4) = 0 \)[/tex], the function touches the [tex]\( x \)[/tex]-axis at this point without crossing it. If [tex]\( g'(-4) \neq 0 \)[/tex], the function crosses the [tex]\( x \)[/tex]-axis at this point.
By substituting [tex]\( x = -4 \)[/tex] into the first derivative, we find that [tex]\( g'(-4) = 0 \)[/tex]. Therefore, the graph touches the [tex]\( x \)[/tex]-axis at [tex]\( x = -4 \)[/tex].
2. Behavior at [tex]\( x = 9 \)[/tex]:
Next, we analyze what happens at [tex]\( x = 9 \)[/tex].
- We know that [tex]\( g(9) = 0 \)[/tex], so [tex]\((9 + 4)^4 (9 - 9) = 0\)[/tex]. This confirms that [tex]\( 9 \)[/tex] is a root of the function.
- Check the value of the first derivative [tex]\( g'(9) \)[/tex]. If [tex]\( g'(9) = 0 \)[/tex], the function touches the [tex]\( x \)[/tex]-axis at this point without crossing it. If [tex]\( g'(9) \neq 0 \)[/tex], the function crosses the [tex]\( x \)[/tex]-axis at this point.
By substituting [tex]\( x = 9 \)[/tex] into the first derivative, we find that [tex]\( g'(9) = 28561 \)[/tex]. Since [tex]\( g'(9) \neq 0 \)[/tex], the graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 9 \)[/tex].
Thus, the detailed analysis of the behavior is as follows:
- At [tex]\( x = -4 \)[/tex], the graph touches the [tex]\( x \)[/tex]-axis.
- At [tex]\( x = 9 \)[/tex], the graph crosses the [tex]\( x \)[/tex]-axis.
