Relative extrema

What you’ll learn

• How to find critical points of various functions
• How to use the 1st derivative test and sign charts to classify critical points as relative maxima, minima, or neither

In the previous section, you learned how to find the highest and lowest values of a function on a closed interval using the Extreme value theorem, which requires comparing function values at

• the endpoints of the interval, and
• any critical points inside the interval.

Now, with the 1st derivative test, a critical point can be classified as a relative (local) maximum, minimum, or neither by analyzing the sign of the derivative, without directly comparing function values.

Absolute vs. relative extrema

An absolute extremum is the highest or lowest value a function attains over its entire domain.

For example, consider $f(x)=x^2$.

The vertex $(0,0)$ is the absolute minimum, or the lowest point on the entire domain of all real numbers. It is also classified as a relative minimum because it’s lower than the points on either side of it.

On the other hand, $x^2$ has no absolute maximum on its full domain, because as $x\to \pm \infty$, $x^2\to \infty$.

Note that the 1st derivative test only identifies relative and not absolute extrema.

First derivative test

Use this process to locate and classify the relative extrema of a function $f(x)$:

Step 1: Find the critical points

Identify all values of $x$ where

• $f(x)=0$

• $f^{\prime }(x)$ is undefined

Note that $x=c$ is only a critical point if it lies within the domain of $f(x)$ (meaning $f(c)$ must exist). The next page will discuss this in more detail.

Step 2: Create a sign diagram or chart for $f^{\prime }(x)$

Place the critical points on a number line to divide the domain into intervals. Choose a test value within each interval and plug it into $f^{\prime }(x)$ to determine whether the derivative is positive or negative.

Step 3: Interpret each critical point

Analyze the behavior of $f^{\prime }(x)$ across each critical point $x=c$:

• Relative maximum if $f^{\prime }(x)$ changes from positive to negative.

  • This means $f(x)$ increases $(\nearrow )$ and then decreases $(\searrow )$.

• Relative minimum if $f^{\prime }(x)$ changes from negative to positive.

  • This means $f(x)$ decreases $(\searrow )$ and then increases $(\nearrow )$.

• No extremum if $f^{\prime }(x)$ does not change sign, meaning $f(x)$ is flat at $x=c$ but continues in the same direction.

  • e.g. $y=x^3$ is flat at $(0,0)$ but the function continues increasing through that point.

AP tip:

Justification of extrema on the AP exam must include statements about explicit derivative behavior. For example,

• Correct: “$f(x)$ has a relative maximum at $x=1$ because $f^{\prime }(x)$ changes from positive to negative.”

• Incorrect: “There is a maximum because the graph goes up and then down.”

  • Too vague and does not name $f$ or $f^{\prime }$

Example 1: $f^{\prime }(c)=0$

Classify the relative extrema of

$$f(x)=(x-2{)}^2(x+1{)}^3$$

Step 1: Find critical points

Step 2: Create a sign chart

Step 3: Interpret

Example 2: $f^{\prime }(c)$ is undefined

Classify the extrema on

$$f(x)=x^{2/3}(x-5)$$

Step 1: Critical points

Step 2: Sign chart

Step 3: Interpret

Increasing/decreasing intervals

The 1st derivative test works because the sign of $f^{\prime }(x)$ indicates the behavior of $f(x)$:

• $f$ is increasing on an interval where $f^{\prime }(x)>0$.
• $f$ is decreasing on an interval where $f^{\prime }(x)<0$.

Many AP problems will ask you to find these intervals. The calculus process is identical: find the critical points, set up a sign chart, and identify the regions.

Let $f(x)=e^{\sin (x)}$ on the open interval $(0,2\pi )$. Find the intervals on which $f$ is increasing.

Step 1: Find critical points

Step 2: Sign chart

Step 3: Interpret