Problem 120 - University Calculus: Early Transcendentals

Question

Suppose that the second derivative of the function \(y=f(x)\) is $$y^{\prime \prime}=x^{2}(x-2)^{3}(x+3).$$ For what \(x\) -values does the graph of \(f\) have an inflection point?

Step-by-Step Solution

Answer

The inflection points are at \( x = -3 \) and \( x = 2 \).

Step 1: Set the Second Derivative to Zero

An inflection point occurs where the concavity of a function changes, which is when the second derivative changes sign. To find these points, set the second derivative equal to zero: \( y'' = x^2 (x-2)^3 (x+3) = 0 \).

Step 2: Solve for Critical Points

Solve the equation from Step 1 for \( x \):1. \( x^2 = 0 \) gives \( x = 0 \).2. \( (x-2)^3 = 0 \) gives \( x = 2 \).3. \( (x+3) = 0 \) gives \( x = -3 \).Thus, the critical points are \( x = 0, 2, \) and \( -3 \).

Step 3: Test Intervals Around Critical Points

Identify intervals around the critical points \( x = -3, 0, \) and \( 2 \) to determine where the concavity changes. Use test points within the intervals to evaluate the sign of \( y'' \):1. For \( x < -3 \), for example \( x = -4 \): \( (-4)^2((-4-2)^3)(-4+3) = 1(negative)(negative) = positive \).2. Between \( x = -3 \) and \( x = 0 \), use \( x = -1 \): \( (-1)^2((-1-2)^3)(-1+3) = 1(negative)(positive) = negative \).3. Between \( x = 0 \) and \( x = 2 \), use \( x = 1 \): \( 1^2(1-2)^3(1+3) = 1(negative)(positive) = negative \).4. For \( x > 2 \), use \( x = 3 \): \( 3^2(3-2)^3(3+3) = 1(positive)(positive) = positive \).

Step 4: Identify Change of Concavity

The concavity changes at \( x = -3, 0, \) and \( 2 \) based on the sign changes from your test intervals: 1. From positive to negative at \( x = -3 \), 2. From negative to negative at \( x = 0 \) (no change of concavity), 3. From negative to positive at \( x = 2 \).

Step 5: Conclude Inflection Points

Since an inflection point occurs where the second derivative changes sign, \( x = -3 \) and \( x = 2 \) are the inflection points. At \( x = 0 \), the concavity does not change, hence it is not an inflection point.

Key Concepts

Second Derivative

The second derivative of a function provides insight into its concavity and helps locate inflection points. When analyzing a function, we often compute its second derivative to examine how the graph bends or "curves." If the second derivative is positive, the graph is concave up, resembling a U-shape. If negative, the graph is concave down, similar to an upside-down U. For finding inflection points, set the second derivative to zero and solve for the variable. This provides potential locations where the graph might change its concavity. Checking these points carefully is crucial, as setting the second derivative to zero gives critical points, which includes potential inflection points for concavity changes.

Concavity Change

A change in the concavity of a graph marks the location of an inflection point. This occurs when the second derivative changes its sign. To identify whether such a change happens at a point, test values around the critical points (obtained from setting the second derivative to zero). For example, suppose we have intervals defined by these points. We select test values within these intervals and substitute them back into the second derivative equation to check for sign change. A switch from a positive to a negative sign, or vice versa, indicates that the concavity indeed changes at that point, confirming the presence of an inflection point.

Critical Points

In the quest to find inflection points, understanding critical points is key. Critical points arise when the first or second derivative of a function is zero or undefined. For this exercise, our focus is primarily on points from the second derivative set to zero. They act as potential markers for inflection by suggesting where change might occur. Solving the equation gives several values of x - these are the critical points. However, not all such critical points will be inflection points, hence further investigation is necessary to ascertain changes in concavity.

Graph Analysis

Analyzing the graph of a function visually confirms the behavior predicted by the derivatives. By plotting or sketching the curve, we can observe changes in direction or curvature. The critical points pinpointed through mathematical calculations translate into visual clues on the graph. For example, an inflection point signifies where the graph alters its bending direction - from concave up to concave down, or vice versa. Beyond mere calculation, analyzing graphs equipped with this derivative information encapsulates the dynamics of how functions behave visually, offering a concrete interpretation of abstract mathematical concepts.