Determining Concavity of Functions over Their Domains - AP Calculus BC Study Guide

The Core Idea: Determining Concavity of Functions over Their Domains

In calculus, the first derivative, $f^{\prime }(x)$, describes a function's rate of change, allowing us to determine where the function is increasing or decreasing. The second derivative, $f^{\prime \prime }(x)$, takes this analysis a step further by describing the rate of change of the first derivative. This reveals the concavity of the original function's graph—whether the graph is curved upwards (like a cup holding water) or downwards (like a cup spilling water). By analyzing the sign of the second derivative, we can precisely identify the intervals over which a function is "concave up" or "concave down."

This topic also introduces the concept of a point of inflection, which is a specific point on the graph where the concavity changes from up to down, or vice versa. These points are critical for understanding the complete shape and behavior of a function's graph. Finding these points involves identifying where the second derivative is equal to zero or is undefined, as these are the only locations where a change in concavity can occur.

Key Definitions and Tests

The Second Derivative

The second derivative of a function $f$, denoted $f^{\prime \prime }(x)$, is the derivative of the first derivative, $f^{\prime }(x)$.

$$f^{\prime \prime }(x)=\frac{d}{dx}\left[f^{\prime }(x)\right]$$

The Concavity Test

The sign of the second derivative on an interval determines the concavity of the graph of $f(x)$ on that same interval.

• If $f^{\prime \prime }(x)>0$ for all $x$ in an interval $I$, then the graph of $f(x)$ is concave up on $I$.

• If $f^{\prime \prime }(x)<0$ for all $x$ in an interval $I$, then the graph of $f(x)$ is concave down on $I$.

Point of Inflection

A point of inflection is a point on the graph of a function $f(x)$ at which the concavity of the graph changes.

• For a point $(c,f(c))$ to be a point of inflection, the function must change from concave up to concave down, or from concave down to concave up, at $x=c$.

• A necessary condition for $(c,f(c))$ to be a point of inflection is that either $f^{\prime \prime }(c)=0$ or $f^{\prime \prime }(c)$ is undefined. These values of $c$ are considered potential points of inflection and must be tested for a change in concavity.

Understanding the Relationship Between $f$, $f^{\prime }$, and $f^{\prime \prime }$

The core of this topic is understanding the chain of relationships between a function and its first two derivatives.

• The sign of $f^{\prime }(x)$ tells you if $f(x)$ is increasing or decreasing.

• The sign of $f^{\prime \prime }(x)$ tells you if $f^{\prime }(x)$ is increasing or decreasing.

This second point is the key to understanding concavity.

• Concave Up: When $f^{\prime \prime }(x)>0$, it means that $f^{\prime }(x)$ (the slope of $f(x)$) is increasing. As you move from left to right on the graph of $f(x)$, the tangent line slopes are becoming less negative, or more positive. This causes the graph of $f(x)$ to bend upwards.

• Concave Down: When $f^{\prime \prime }(x)<0$, it means that $f^{\prime }(x)$ (the slope of $f(x)$) is decreasing. As you move from left to right on the graph of $f(x)$, the tangent line slopes are becoming more negative, or less positive. This causes the graph of $f(x)$ to bend downwards.

A critical nuance involves points of inflection. The condition $f^{\prime \prime }(c)=0$ or $f^{\prime \prime }(c)$ being undefined does not guarantee a point of inflection. It only provides a list of candidates. The definitive test is whether the sign of $f^{\prime \prime }(x)$ changes around $x=c$. If $f^{\prime \prime }(x)$ is positive on both sides of $c$, for example, then the function is concave up on both sides, and no change in concavity occurs at $c$.

Core Concepts & Rules

• The second derivative, $f^{\prime \prime }(x)$, is the derivative of the first derivative, $f^{\prime }(x)$.

• The sign of $f^{\prime \prime }(x)$ determines the concavity of the graph of $f(x)$.

• If $f^{\prime \prime }(x)>0$ on an interval, the graph of $f(x)$ is concave up on that interval. This also means that $f^{\prime }(x)$ is increasing on that interval.

• If $f^{\prime \prime }(x)<0$ on an interval, the graph of $f(x)$ is concave down on that interval. This also means that $f^{\prime }(x)$ is decreasing on that interval.

• A point of inflection is a point on the graph of $f(x)$ where the concavity changes.

• To find points of inflection, first identify all values $c$ in the domain of $f(x)$ where $f^{\prime \prime }(c)=0$ or $f^{\prime \prime }(c)$ is undefined. These are the potential points of inflection.

• A point of inflection exists at $x=c$ only if the sign of $f^{\prime \prime }(x)$ changes at $x=c$.

Step-by-Step Example 1: Analytical Application

Problem: Find the intervals of concavity and the coordinates of any points of inflection for the function $f(x)=x^4-6x^2+3$.

Step 1: Find the first derivative, $f^{\prime }(x)$.

$$f^{\prime }(x)=4x^3-12x$$

Step 2: Find the second derivative, $f^{\prime \prime }(x)$.

$$f^{\prime \prime }(x)=12x^2-12$$

Step 3: Find potential points of inflection.

Set $f^{\prime \prime }(x)=0$ and solve for $x$. Note that $f^{\prime \prime }(x)$ is a polynomial and is never undefined.

$$12x^2-12=0$$

$$12(x^2-1)=0$$

$$12(x-1)(x+1)=0$$

The potential points of inflection occur at $x=-1$ and $x=1$.

Step 4: Create a sign chart for $f^{\prime \prime }(x)$.

Use the potential points of inflection to create intervals for testing. The intervals are $(-\infty ,-1)$, $(-1,1)$, and $(1,\infty )$.

Step 5: State the conclusions.

• The graph of $f(x)$ is concave up on the intervals $(-\infty ,-1)$ and $(1,\infty )$ because $f^{\prime \prime }(x)>0$ on these intervals.

• The graph of $f(x)$ is concave down on the interval $(-1,1)$ because $f^{\prime \prime }(x)<0$ on this interval.

Step 6: Find the coordinates of the points of inflection.

Since the concavity changes at both $x=-1$ and $x=1$, both are points of inflection. Find the y-coordinates by plugging these x-values back into the original function, $f(x)$.

• For $x=-1$: $f(-1)=(-1{)}^4-6(-1{)}^2+3=1-6+3=-2$.

• For $x=1$: $f(1)=(1{)}^4-6(1{)}^2+3=1-6+3=-2$.

The points of inflection are $(-1,-2)$ and $(1,-2)$.

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

Problem: The graph of $f^{\prime }(x)$, the derivative of a continuous function $f(x)$, is shown below. Use the graph to determine the intervals on which $f(x)$ is concave up and concave down, and identify the x-coordinates of any points of inflection.

(Imagine a graph of $f^{\prime }(x)$ that is a parabola opening downwards, with x-intercepts at $x=-2$ and $x=4$, and a vertex at $(1,3)$.)

Step 1: Relate the concavity of $f(x)$ to the given graph of $f^{\prime }(x)$.

• The concavity of $f(x)$ is determined by the sign of $f^{\prime \prime }(x)$.

• The second derivative, $f^{\prime \prime }(x)$, is the derivative of $f^{\prime }(x)$.

• Therefore, the sign of $f^{\prime \prime }(x)$ is determined by the slope of the given graph of $f^{\prime }(x)$.

• $f(x)$ is concave up where $f^{\prime \prime }(x)>0$, which means where $f^{\prime }(x)$ is increasing.

• $f(x)$ is concave down where $f^{\prime \prime }(x)<0$, which means where $f^{\prime }(x)$ is decreasing.

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

• By observing the provided graph, the function $f^{\prime }(x)$ is increasing on the interval $(-\infty ,1)$.

• The function $f^{\prime }(x)$ is decreasing on the interval $(1,\infty )$.

Step 3: Determine the intervals of concavity for $f(x)$.

• Because $f^{\prime }(x)$ is increasing on $(-\infty ,1)$, we conclude that $f^{\prime \prime }(x)>0$ on this interval. Therefore, $f(x)$ is concave up on $(-\infty ,1)$.

• Because $f^{\prime }(x)$ is decreasing on $(1,\infty )$, we conclude that $f^{\prime \prime }(x)<0$ on this interval. Therefore, $f(x)$ is concave down on $(1,\infty )$.

Step 4: Identify the points of inflection.

• A point of inflection occurs where the concavity of $f(x)$ changes.

• This change occurs at $x=1$, where the slope of $f^{\prime }(x)$ changes from positive to negative. At this point, the derivative of $f^{\prime }(x)$ (which is $f^{\prime \prime }(x)$) is zero.

• Therefore, $f(x)$ has a point of inflection at $x=1$.

Using Your Calculator

The process of determining concavity is primarily analytical. However, a graphing calculator can be used to verify results or analyze functions that are difficult to differentiate by hand.

To find concavity and points of inflection for $f(x)$:

1. Enter the function $f(x)$ into Y1.

2. To analyze the second derivative, you can graph it in Y2. Use the calculator's numerical derivative feature twice.

   • In the Y= editor, for Y2, use the math template for the derivative: $d/dx(...)$.

   • Inside the template, place another $d/dx(...)$ template.

   • The expression should look like: Y2 = d/dx(d/dx(Y1, X, X), X, X) or use the $d^2/d{x}^2$ template if available.

3. Graph Y2 (the graph of $f^{\prime \prime }(x)$).

4. Determine Concavity:

   • Find the intervals where the graph of Y2 is above the x-axis. On these intervals, $f^{\prime \prime }(x)>0$, so $f(x)$ is concave up.

   • Find the intervals where the graph of Y2 is below the x-axis. On these intervals, $f^{\prime \prime }(x)<0$, so $f(x)$ is concave down.

5. Find Points of Inflection:

   • Use the calculator's "zero" or "root" finding feature on the graph of Y2 to find the x-values where $f^{\prime \prime }(x)=0$.

   • Visually confirm that the graph of Y2 crosses the x-axis at these points (i.e., the sign of $f^{\prime \prime }(x)$ changes). These are the x-coordinates of the points of inflection.

AP Exam Quick Hit

Common Question Types

• Analytical from an Equation: Given a function $f(x)$, find the intervals of concavity and the coordinates of the points of inflection. This requires finding $f^{\prime \prime }(x)$, setting it to zero, and using a sign chart.

• Graphical from $f^{\prime }(x)$ or $f^{\prime \prime }(x)$: Given the graph of $f^{\prime }(x)$, determine the concavity of $f(x)$ by analyzing the intervals where $f^{\prime }(x)$ is increasing or decreasing. Given the graph of $f^{\prime \prime }(x)$, determine the concavity of $f(x)$ by analyzing the intervals where $f^{\prime \prime }(x)$ is positive or negative.

• Tabular Data: Given a table of values for $x$, $f^{\prime }(x)$, and $f^{\prime \prime }(x)$, justify why $f(x)$ is concave up or down on an interval. For example, if all values of $f^{\prime \prime }(x)$ in the table for the interval $[a,b]$ are positive, you can conclude $f(x)$ is concave up on that interval.

Common Mistakes

• Confusing $f^{\prime }$ and $f^{\prime \prime }$: Stating that $f(x)$ is concave up because $f(x)$ or $f^{\prime }(x)$ is positive. Concavity is determined only by the sign of $f^{\prime \prime }(x)$.

• Assuming Inflection Point at $f^{\prime \prime }(c)=0$: A student finds where $f^{\prime \prime }(c)=0$ but fails to test whether the sign of $f^{\prime \prime }(x)$ actually changes at $x=c$. For $f(x)=x^4$, $f^{\prime \prime }(0)=0$, but there is no point of inflection because the graph is concave up on both sides of $x=0$.

• Incorrect y-coordinate for Inflection Point: After finding the x-coordinate $c$ of an inflection point, plugging it into $f^{\prime }(x)$ or $f^{\prime \prime }(x)$ to find the y-coordinate. The point of inflection $(c,f(c))$ lies on the graph of the original function $f(x)$.

• Misinterpreting the Graph of $f^{\prime }(x)$: When given the graph of $f^{\prime }(x)$, looking at the concavity of the $f^{\prime }(x)$ graph itself to determine the concavity of $f(x)$. You must analyze the slope of the $f^{\prime }(x)$ graph, as the slope of $f^{\prime }(x)$ is $f^{\prime \prime }(x)$.