Using the First Derivative Test to Determine Relative (Local) Extrema - AP Calculus AB Study Guide

The Core Idea: Using the First Derivative Test to Determine Relative (Local) Extrema

The first derivative of a function, $f^{\prime }(x)$, describes the function's rate of change, or its slope, at any given point. This means the sign of $f^{\prime }(x)$ tells us whether the original function $f(x)$ is increasing (where $f^{\prime }(x)>0$) or decreasing (where $f^{\prime }(x)<0$). The First Derivative Test leverages this relationship to analyze the behavior of a function at its critical points—the points where $f^{\prime }(x)=0$ or is undefined.

By examining how the sign of the first derivative changes around a critical point, we can determine if that point corresponds to a relative (local) maximum (a "peak"), a relative (local) minimum (a "valley"), or neither. If the function's slope changes from positive to negative, it has reached a peak. If the slope changes from negative to positive, it has bottomed out in a valley. This test provides a systematic method for classifying the "hills and valleys" on the graph of a function.

The First Derivative Test

The First Derivative Test is a procedure used to classify a critical point of a function $f(x)$ as a relative maximum, relative minimum, or neither. Let $x=c$ be a critical point of the function $f(x)$.

1. Relative (Local) Maximum: If $f^{\prime }(x)$ changes from positive to negative at $x=c$, then $f(x)$ has a relative maximum at $x=c$.

   • This means the function $f(x)$ was increasing to the left of $c$ and is decreasing to the right of $c$.

2. Relative (Local) Minimum: If $f^{\prime }(x)$ changes from negative to positive at $x=c$, then $f(x)$ has a relative minimum at $x=c$.

   • This means the function $f(x)$ was decreasing to the left of $c$ and is increasing to the right of $c$.

3. Neither a Maximum nor a Minimum: If $f^{\prime }(x)$ does not change sign at $x=c$ (i.e., it is positive on both sides of $c$ or negative on both sides of $c$), then $f(x)$ has neither a relative maximum nor a relative minimum at $x=c$.

Understanding Critical Points

A critical point is a necessary condition for a relative extremum to exist, but it is not a guarantee. The First Derivative Test is the tool that confirms whether a candidate critical point is actually an extremum.

The core of the test is the link between the sign of $f^{\prime }(x)$ and the behavior of $f(x)$.

• When $f^{\prime }(x)>0$, the function $f(x)$ is increasing.

• When $f^{\prime }(x)<0$, the function $f(x)$ is decreasing.

Consider a critical point $x=c$.

• Maximum: To have a peak at $x=c$, the function must go from increasing to decreasing. This corresponds to its derivative, $f^{\prime }(x)$, changing from positive values to negative values.

• Minimum: To have a valley at $x=c$, the function must go from decreasing to increasing. This corresponds to its derivative, $f^{\prime }(x)$, changing from negative values to positive values.

• Neither: If the function is increasing, flattens out momentarily at $x=c$, and then continues increasing, there is no extremum. This corresponds to $f^{\prime }(x)$ being positive, becoming zero at $x=c$, and then being positive again. The sign of $f^{\prime }(x)$ does not change. A classic example is $f(x)=x^3$ at $x=0$.

Core Concepts & Rules

• The First Derivative Test is used to classify critical points of a function as relative maxima, relative minima, or neither.

• A relative maximum occurs at a critical point $x=c$ if the sign of $f^{\prime }(x)$ changes from positive to negative at $x=c$.

• A relative minimum occurs at a critical point $x=c$ if the sign of $f^{\prime }(x)$ changes from negative to positive at $x=c$.

• If the sign of $f^{\prime }(x)$ is the same on both sides of a critical point $x=c$, then the function has neither a relative maximum nor a relative minimum at that point.

• The justification for a relative extremum must be based on the sign change of the first derivative.

Step-by-Step Example 1: Analytical Application

Find and classify all relative extrema for the function $f(x)=\frac{1}{3}{x}^3-x^2-8x+1$.

Step 1: Find the first derivative of the function.

The derivative gives us information about the slope of $f(x)$.

$$f^{\prime }(x)=\frac{d}{dx}\left(\frac{1}{3}{x}^3-x^2-8x+1\right)=x^2-2x-8$$

Step 2: Find the critical points.

Critical points occur where $f^{\prime }(x)=0$ or $f^{\prime }(x)$ is undefined. Since $f^{\prime }(x)$ is a polynomial, it is defined for all real numbers. We only need to find where $f^{\prime }(x)=0$.

$$x^2-2x-8=0$$

$$(x-4)(x+2)=0$$

The critical points are $x=4$ and $x=-2$.

Step 3: Create a sign chart for $f^{\prime }(x)$.

Draw a number line and mark the critical points. These points divide the number line into three intervals: $(-\infty ,-2)$, $(-2,4)$, and $(4,\infty )$.

Step 4: Test the sign of $f^{\prime }(x)$ in each interval.

Pick a test value within each interval and substitute it into the factored form of the derivative, $f^{\prime }(x)=(x-4)(x+2)$, to determine its sign.

• Interval $(-\infty ,-2)$: Let's test $x=-3$.

  $f^{\prime }(-3)=(-3-4)(-3+2)=(-7)(-1)=7$. The sign is positive.

• Interval $(-2,4)$: Let's test $x=0$.

  $f^{\prime }(0)=(0-4)(0+2)=(-4)(2)=-8$. The sign is negative.

• Interval $(4,\infty )$: Let's test $x=5$.

  $f^{\prime }(5)=(5-4)(5+2)=(1)(7)=7$. The sign is positive.

Step 5: Apply the First Derivative Test and state the conclusion.

Analyze the sign changes at each critical point.

• At $x=-2$, the sign of $f^{\prime }(x)$ changes from positive to negative. Therefore, $f(x)$ has a relative maximum at $x=-2$.

• At $x=4$, the sign of $f^{\prime }(x)$ changes from negative to positive. Therefore, $f(x)$ has a relative minimum at $x=4$.

Step-by-Step Example 2: Exam-Style Application (Graphical)

The graph of $f^{\prime }(x)$, the derivative of a continuous function $f(x)$, is shown below on the interval $[-4,5]$. Find the x-coordinates of all relative extrema of $f(x)$ and classify them. Justify your answer.

(Imagine a graph of $f^{\prime }(x)$ that starts below the x-axis, crosses up at $x=-3$, touches the x-axis at $x=1$, and crosses down at $x=4$.)

Step 1: Identify the critical points of $f(x)$ from the graph of $f^{\prime }(x)$.

Critical points of $f(x)$ occur where $f^{\prime }(x)=0$. Looking at the provided graph, this happens when the graph of $f^{\prime }(x)$ intersects or touches the x-axis.

The critical points are $x=-3$, $x=1$, and $x=4$.

Step 2: Analyze the sign of $f^{\prime }(x)$ around each critical point.

The "sign" of $f^{\prime }(x)$ corresponds to whether its graph is above or below the x-axis.

• Around $x=-3$: To the left of $x=-3$, the graph of $f^{\prime }(x)$ is below the x-axis ($f^{\prime }(x)<0$). To the right of $x=-3$, the graph is above the x-axis ($f^{\prime }(x)>0$).

• Around $x=1$: To the left of $x=1$, the graph of $f^{\prime }(x)$ is above the x-axis ($f^{\prime }(x)>0$). To the right of $x=1$, the graph is also above the x-axis ($f^{\prime }(x)>0$).

• Around $x=4$: To the left of $x=4$, the graph of $f^{\prime }(x)$ is above the x-axis ($f^{\prime }(x)>0$). To the right of $x=4$, the graph is below the x-axis ($f^{\prime }(x)<0$).

Step 3: Apply the First Derivative Test and write the justification.

Use the sign changes identified in Step 2 to classify each critical point.

• At $x=-3$:$f(x)$ has a relative minimum because $f^{\prime }(x)$ changes from negative to positive at $x=-3$.

• At $x=1$:$f(x)$ has neither a relative maximum nor a minimum because $f^{\prime }(x)$ does not change sign at $x=1$.

• At $x=4$:$f(x)$ has a relative maximum because $f^{\prime }(x)$ changes from positive to negative at $x=4$.

Using Your Calculator

While the First Derivative Test is an analytical method, a graphing calculator can be used to support your work, especially for complex functions. The primary use is to visualize the derivative to quickly identify its sign.

To find and classify extrema of $f(x)$:

1. Graph the Derivative, $f^{\prime }(x)$:

   • If you have the function $f(x)$, you can graph its derivative directly in the Y= screen. Let $Y_1=f(x)$. In $Y_2$, use the numerical derivative feature. For a TI-84 style calculator, this is nDeriv(Y_1, X, X).

   • Y₂ = nDeriv(Y₁, X, X)

   • Graph $Y_2$. Be sure to turn off the graph of $Y_1$ so you are only looking at the derivative.

2. Analyze the Graph of $f^{\prime }(x)$:

   • Use the CALC menu's $zero$ feature to find the x-intercepts of the $f^{\prime }(x)$ graph. These are the critical points of $f(x)$.

   • Visually inspect the graph of $f^{\prime }(x)$ at each zero (critical point):

     • If the graph of $f^{\prime }(x)$crosses from below to above the x-axis, the sign of $f^{\prime }(x)$ changes from negative to positive. This indicates a relative minimum for $f(x)$.

     • If the graph of $f^{\prime }(x)$crosses from above to below the x-axis, the sign of $f^{\prime }(x)$ changes from positive to negative. This indicates a relative maximum for $f(x)$.

     • If the graph of $f^{\prime }(x)$touches but does not cross the x-axis, the sign of $f^{\prime }(x)$ does not change. This indicates neither a max nor a min for $f(x)$.

Important Note: Your written justification on the AP Exam must be based on the sign change of the derivative (e.g., "$f(x)$ has a relative max at $x=c$ because $f^{\prime }(x)$ changes from positive to negative at $x=c$"), not on the visual appearance of the graph.

AP Exam Quick Hit

Common Question Types

• Analytical Function: You are given an equation for $f(x)$. You must find $f^{\prime }(x)$, find the critical points by setting $f^{\prime }(x)=0$, and use a sign chart to classify the extrema.

• Graphical Derivative: You are given the graph of $f^{\prime }(x)$ (as in Example 2). You must identify the x-intercepts of the graph as critical points and classify them based on where the graph is above (positive) or below (negative) the x-axis.

• Tabular Derivative: You are given a table of values for $f^{\prime }(x)$. You might be told that $f^{\prime }(c)=0$ and asked to determine if $f(x)$ has a relative extremum at $x=c$. You would look at the signs of $f^{\prime }(x)$ for values in the table just before and just after $c$.

Common Mistakes

• Confusing $f(x)$ with $f^{\prime }(x)$: When given the graph of $f^{\prime }(x)$, students mistakenly identify the peaks and valleys on the graph of $f^{\prime }(x)$ as the extrema of $f(x)$. Remember, you must look at where the graph of $f^{\prime }(x)$ crosses the x-axis.

• Incomplete Justification: Stating that $f(x)$ has a maximum because "the function changes from increasing to decreasing." While true, this is not a complete justification. The required justification must be in terms of the derivative: "$f(x)$ has a relative maximum because $f^{\prime }(x)$ changes from positive to negative."

• Stopping After Finding Critical Points: A common error is to find where $f^{\prime }(x)=0$ and assume these are all extrema. You must test the sign of $f^{\prime }(x)$ on either side of each critical point to classify it.

• Forgetting "Neither": Assuming every critical point must be a maximum or a minimum. If the derivative does not change sign (e.g., positive on both sides), the point is not a relative extremum.

• Mixing up Max and Min: Confusing the sign changes. A helpful mnemonic: a positive to negative change is a Peak (Maximum), and a negative to positive change is a Pit (Minimum).