Functions and Graphs
2.4 Maxima, Minima, and Rate of Change
Behavior Over Intervals
Instead of a single point value, we now ask how a function behaves as $x$ changes.
Definition: Increasing and Decreasing
A function is increasing on an interval if larger $x$ gives larger $f(x)$.
A function is decreasing on an interval if larger $x$ gives smaller $f(x)$.
Definition: Local Extrema
A local maximum is a high point relative to nearby points.
A local minimum is a low point relative to nearby points.
Example: Read extrema from vertex form
For
$$f(x)=-(x-2)^2+5,$$
identify local extrema and increasing/decreasing intervals.
Show Solution
This parabola opens downward.
Vertex is $(2,5)$, so local maximum is $5$ at $x=2$.
It is increasing on $(-\infty,2)$ and decreasing on $(2,\infty)$.
MyOpenMath: Practice: intervals of increase/decrease (placeholder)
Average Rate of Change
Average rate of change is slope between two points on a graph.
Definition: Average Rate of Change
For $x_1\ne x_2$,
$$\text{AROC} = \frac{f(x_2)-f(x_1)}{x_2-x_1}. $$
Example: Compute average rate of change
For $f(x)=x^2-4x$, find average rate of change from $x=1$ to $x=4$.
Show Solution
Compute outputs:
$$f(1)=1-4=-3,$$ $$f(4)=16-16=0.$$
Then
$$\frac{f(4)-f(1)}{4-1}=\frac{0-(-3)}{3}=1.$$
Average rate of change is $1$.