5.5 Low Points and High Points
Let's return to the problem of locating a graph's low points and high points. First, we should locate the graph's level points.
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\slope}} z &= 0
\end{align}
\]
Now that we can calculate the bend, we can use it to help decide whether each level point is a low point or a high point.
Important.
Suppose we've found a level point for a function \( z \depends x \).
If the graph has positive bend at the level point,
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &\gt 0,
\end{align}
\]
then that is a low point for the graph.
If the graph has negative bend at the level point,
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &\lt 0,
\end{align}
\]
then that is a high point for the graph.
If a level point has zero bend, then the level point might be a low point, a high point, or an inflection point! In this case, we might draw a transition diagram for slope to better understand the shape of the graph.
Example.
Let's analyze the level points for the function
\[
\begin{align}
z &\depends x
\\
z &= x^3 - 3 x.
\end{align}
\]
We looked at this function once before in the section 4.5 Analyzing Slope. By calculating the slope
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\slope}} z &= 3 x^2 - 3,
\end{align}
\]
we found that this function has two level points: \( x = -1 \) and \( x = 1 \). Let's use the bend
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &= 3 x
\end{align}
\]
to classify the level points.
At \( x = -1 \).
Localizing the bend, we find
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &= -3.
\end{align}
\]
A negative bend means the graph is concave down, and so \( x = -1 \) is a high point.
At \( x = 1 \).
Here the bend is positive.
\[
\begin{align}
\operatorname{\overset{\mathit{x}}{\bend}} z &= 3
\end{align}
\]
The graph is concave up, and so \( x = 1 \) must be a low point.
Let's conclude by making a table of the height, slope, and bend at each of the level points.
| \( \vphantom{\overset{x}{o}} x \) | \( \vphantom{\overset{x}{o}} z \) | \( \operatorname{\overset{\mathit{x}}{\slope}} z \) | \( \operatorname{\overset{\mathit{x}}{\bend}} z \) |
|---|---|---|---|
| \( -1 \) | \( 2 \) | \( 0 \) | \( -3 \) |
| \( 1 \) | \( -2 \) | \( 0 \) | \( 3 \) |
The graph has height \( z = 2 \) at the high point, and height \( z = -2 \) at the low point.