Determining Intervals on Which a Function Is Increasing or Decreasing - AP Calculus BC Study Guide

The Core Idea: Determining Intervals on Which a Function Is Increasing or Decreasing

The first derivative of a function, $f^{\prime }(x)$, provides powerful insight into the behavior of the original function, $f(x)$. At its core, the derivative represents the slope of the tangent line to the function at any given point. By analyzing the sign (positive or negative) of the first derivative across different intervals of the domain, we can determine precisely where the original function is rising (increasing) or falling (decreasing).

This topic establishes the fundamental connection between a function and its derivative. The process involves finding "critical points"—locations where the function's behavior might change—and then testing the sign of the derivative in the intervals separated by these points. A positive derivative signifies a positive slope, meaning the function is increasing. Conversely, a negative derivative signifies a negative slope, meaning the function is decreasing.

Key Definitions and Rules

The relationship between the sign of the first derivative and the behavior of the function is governed by the following rules and definitions.

• Condition for an Increasing Function: A function $f$ is increasing on an interval if its first derivative is positive for all $x$ in that interval.

  $$f^{\prime }(x)>0⟹f(x)\text{ is increasing}$$

• Condition for a Decreasing Function: A function $f$ is decreasing on an interval if its first derivative is negative for all $x$ in that interval.

  $$f^{\prime }(x)<0⟹f(x)\text{ is decreasing}$$

• Definition of Critical Points: The critical points of a function $f$ are the $x$-values in the domain of $f$ where the derivative $f^{\prime }(x)$ is either equal to zero or is undefined. These points are the only candidates for where a function might change from increasing to decreasing, or vice versa.

  $$\text{Critical Points occur where }{f}^{\prime }(x)=0\text{ or }{f}^{\prime }(x)\text{ is undefined.}$$

Understanding the Connection Between $f^{\prime }$ and $f$

The rules for determining intervals of increasing or decreasing behavior are not arbitrary; they are a direct consequence of the definition of the derivative. The value of $f^{\prime }(c)$ is the instantaneous rate of change, or the slope of the tangent line, of $f(x)$ at $x=c$.

Imagine a function that is increasing. If you were to draw tangent lines at any point in that interval, they would all have a positive slope, pointing upwards from left to right. This visual observation corresponds to the analytical condition $f^{\prime }(x)>0$. Similarly, for a decreasing function, all tangent lines would have a negative slope, pointing downwards from left to right, which corresponds to $f^{\prime }(x)<0$.

The critical points are essential because they are the only points where the slope can change its sign. A slope can only transition from positive to negative (or vice versa) by passing through a slope of zero ($f^{\prime }(x)=0$, a horizontal tangent) or by passing through a point where the slope is undefined (e.g., a sharp corner or a vertical tangent). Therefore, finding these critical points allows us to partition the function's domain into a set of open intervals. Within each of these intervals, the sign of $f^{\prime }(x)$ will be constant, and thus the behavior of $f(x)$ (either increasing or decreasing) will also be constant.

Core Concepts & Rules

• The sign of the first derivative, $f^{\prime }(x)$, dictates whether the function $f(x)$ is increasing or decreasing.

• If $f^{\prime }(x)$ is positive on an open interval, then $f(x)$ is increasing on that interval.

• If $f^{\prime }(x)$ is negative on an open interval, then $f(x)$ is decreasing on that interval.

• To find the intervals, one must first find all critical points of the function $f(x)$.

• Critical points are the $x$-values where $f^{\prime }(x)=0$ or $f^{\prime }(x)$ is undefined. These points serve as the boundaries for the intervals to be tested.

Step-by-Step Example 1: Finding Intervals for a Polynomial Function

Determine the open intervals on which the function $f(x)=2x^3+3x^2-36x+7$ is increasing or decreasing.

Step 1: Find the First Derivative

First, we compute $f^{\prime }(x)$ using the power rule.

$$f^{\prime }(x)=\frac{d}{dx}(2x^3+3x^2-36x+7)$$

$$f^{\prime }(x)=6x^2+6x-36$$

Step 2: Find the Critical Points

Next, we find the $x$-values 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 set $f^{\prime }(x)=0$.

$$6x^2+6x-36=0$$

$$6(x^2+x-6)=0$$

$$6(x+3)(x-2)=0$$

The critical points are $x=-3$ and $x=2$.

Step 3: Create a Sign Chart for $f^{\prime }(x)$

The critical points $x=-3$ and $x=2$ divide the number line into three intervals: $(-\infty ,-3)$, $(-3,2)$, and $(2,\infty )$. We will test a value from each interval to determine the sign of $f^{\prime }(x)$.

Step 4: State the Conclusion with Justification

Using the information from the sign chart, we can state our final answer. It is crucial to provide a justification based on the sign of $f^{\prime }(x)$.

• The function $f(x)$ is increasing on the intervals $(-\infty ,-3)\cup (2,\infty )$ because $f^{\prime }(x)>0$ on these intervals.

• The function $f(x)$ is decreasing on the interval $(-3,2)$ because $f^{\prime }(x)<0$ on this interval.

Step-by-Step Example 2: Analyzing a Function with a Cusp

Determine the open intervals on which the function $g(x)=x^{2/3}(x-5)$ is increasing or decreasing.

Step 1: Find the First Derivative

First, rewrite $g(x)$ as $g(x)=x^{5/3}-5x^{2/3}$. Now, differentiate.

$$g^{\prime }(x)=\frac{5}{3}{x}^{2/3}-5\left(\frac{2}{3}\right)x^{-1/3}$$

$$g^{\prime }(x)=\frac{5}{3}{x}^{2/3}-\frac{10}{3}{x}^{-1/3}$$

Step 2: Simplify the Derivative and Find Critical Points

To analyze the sign of $g^{\prime }(x)$, we combine the terms by factoring and finding a common denominator.

$$g^{\prime }(x)=\frac{5}{3}{x}^{-1/3}(x-2)$$

$$g^{\prime }(x)=\frac{5(x-2)}{3x^{1/3}}$$

Now, we find the critical points.

• $g^{\prime }(x)=0$: This occurs when the numerator is zero.

  $$5(x-2)=0⟹x=2$$

• $g^{\prime }(x)$ is undefined: This occurs when the denominator is zero.

  $$3x^{1/3}=0⟹x=0$$

The critical points are $x=0$ and $x=2$.

Step 3: Create a Sign Chart for $g^{\prime }(x)$

The critical points $x=0$ and $x=2$ divide the number line into three intervals: $(-\infty ,0)$, $(0,2)$, and $(2,\infty )$.

Step 4: State the Conclusion with Justification

• The function $g(x)$ is increasing on the intervals $(-\infty ,0)\cup (2,\infty )$ because $g^{\prime }(x)>0$ on these intervals.

• The function $g(x)$ is decreasing on the interval $(0,2)$ because $g^{\prime }(x)<0$ on this interval.

Using Your Calculator

While the process of determining intervals of increasing and decreasing behavior is analytical, a graphing calculator is an excellent tool for verifying your results.

To check your work for a function $f(x)$:

1. Analytically find $f^{\prime }(x)$ and the critical points first. This is the required work for the exam.

2. Graph the derivative, $f^{\prime }(x)$. Enter your calculated derivative function into your calculator's Y1= editor.

3. Analyze the graph of $f^{\prime }(x)$:

   • The intervals where the graph of $f^{\prime }(x)$ is above the x-axis correspond to where $f^{\prime }(x)>0$. The original function $f(x)$ should be increasing on these intervals.

   • The intervals where the graph of $f^{\prime }(x)$ is below the x-axis correspond to where $f^{\prime }(x)<0$. The original function $f(x)$ should be decreasing on these intervals.

4. Verify Critical Points: Use the calculator's "zero" or "root-finding" feature on the graph of $f^{\prime }(x)$ to find its x-intercepts. These values should match the critical points you found analytically where $f^{\prime }(x)=0$. For critical points where $f^{\prime }(x)$ is undefined (like vertical asymptotes), you can observe this behavior on the graph.

AP Exam Quick Hit

Common Question Types

• From an Equation: Given a function $f(x)$, find the open intervals on which $f$ is increasing.

  • Example: "For $f(x)=x{e}^{-2x}$, find the intervals on which $f$ is decreasing."

• From a Graph of the Derivative: The graph of $f^{\prime }(x)$ is provided. Students must identify the intervals where $f(x)$ is increasing or decreasing.

  • Example: "The graph of $f^{\prime }$, the derivative of $f$, is shown above. On which of the following intervals is $f$ increasing?" (The correct answer would be the intervals where the provided graph is above the x-axis).

• From a Table of Values: A table of values for $f^{\prime }(x)$ is given. Students must draw conclusions about the behavior of $f(x)$.

  • Example: "Selected values of the derivative of a function $g$ are given in the table. Based on the table, on which interval must $g$ be decreasing?" (The correct answer would be an interval where all given values of $g^{\prime }(x)$ are negative).

Common Mistakes

• Confusing the Behavior of $f^{\prime }(x)$ with $f(x)$: When given the graph of $f^{\prime }(x)$, a common mistake is to look at where the graph of $f^{\prime }(x)$ is increasing or decreasing. You must look at where the graph of $f^{\prime }(x)$ is positive (above the x-axis) or negative (below the x-axis).

• Forgetting Critical Points from Undefined Derivatives: Students often find critical points by setting $f^{\prime }(x)=0$ but forget to check for $x$-values where $f^{\prime }(x)$ is undefined (e.g., where a denominator is zero or the argument of a natural log is non-positive).

• Insufficient Justification: On free-response questions, writing "the function is increasing because the sign chart is positive" is not a valid justification. The justification must be linked to the derivative: "$f(x)$ is increasing on $(a,b)$ because $f^{\prime }(x)>0$ on that interval."

• Testing the Wrong Function: After finding critical points from $f^{\prime }(x)$, students sometimes plug test values into the original function $f(x)$ instead of the derivative $f^{\prime }(x)$ to determine the sign.

• Algebraic Errors: Simple errors in differentiation or in solving the equation $f^{\prime }(x)=0$ are the most frequent source of incorrect answers. Always double-check your algebra.