Problem 110 - Thomas Calculus in SI Units

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 function has inflection points at \( x = -3 \) and \( x = 2 \).

Step 1: Understand Inflection Points

An inflection point occurs where the function changes concavity. This happens when the second derivative changes its sign. To find potential inflection points, set the second derivative equal to zero and solve for x-values.

Step 2: Set the Second Derivative to Zero

Given the expression for the second derivative:\[ y'' = x^2(x-2)^3(x+3) \]Set it to zero to find critical points:\[ x^2(x-2)^3(x+3) = 0 \]

Step 3: Solve for x from Factored Equation

Solve for x-values from the factored equation we obtained in Step 2: - Factor: \( x^2 = 0 \), gives solution \( x = 0 \) - Factor: \( (x-2)^3 = 0 \), gives solution \( x = 2 \) - Factor: \( (x+3) = 0 \), gives solution \( x = -3 \)

Step 4: Test Intervals Around Critical Points

Test sign changes in \( y'' \) around the critical points to confirm inflection points:- For \( x = -3 \): Pick test values around -3 and observe sign change in \( y'' \)- For \( x = 0 \): There's no sign change because \( x^2 \ge 0 \) always- For \( x = 2 \): Pick test values around 2 and observe sign change in \( y'' \)- Actual changes in concavity occur at \( x = -3 \) and \( x = 2 \)

Key Concepts

second derivative

The second derivative of a function gives us information about the concavity of the graph of the function. It is essentially the derivative of the first derivative.
In mathematical terms, if we have a function \( y = f(x) \), then the second derivative is denoted as \( y'' = \frac{d^2y}{dx^2} \).
This tells us how the rate of change of the slope (the first derivative) itself is changing over time. The second derivative is particularly useful because it helps us understand:

• Whether the function's graph is curving upwards or downwards.
• If there are points where this curvature, or concavity, changes.

In other words, by analyzing the second derivative, we can locate the inflection points in a function's graph.

concavity change

Concavity change in a function's graph occurs when the curvature shifts from being concave upwards to concave downwards, or vice versa. This transition point is what we refer to as an inflection point.
A graph that is concave up looks like a smiling face, while a concave down graph resembles a frown.
To determine a concavity change, the key indicator is a sign change in the second derivative. Mathematicians use the second derivative test:

• If \( y'' > 0 \), the graph is concave up.
• If \( y'' < 0 \), the graph is concave down.

If \( y'' \) changes sign as \( x \) passes through a particular point, the function has a concavity change, and thus, an inflection point at that point. Always remember: for an inflection point, it is not sufficient for the second derivative to be zero; it must change signs.

critical points

Critical points are values of \( x \) where the derivative of a function is either zero or undefined. These points are essential in calculus because they indicate where a graph might have a local maximum or minimum, or an inflection point.
In the context of the second derivative, critical points are found where the second derivative itself is zero. For our specific example, given the second derivative \( y'' = x^2(x-2)^3(x+3) \):

• We set \( y'' = 0 \)
• This gives us potential inflection points at \( x = 0 \), \( x = 2 \), and \( x = -3 \).

Not all critical points are inflection points, though. We need to test the intervals around these points to identify actual changes in concavity.

calculus problem-solving

Solving calculus problems, such as finding inflection points, involves a systematic approach. Calculus allows us to explore the behavior of functions and make predictions based on derivatives. Here is a basic guide for identifying inflection points:

• Start with finding the second derivative of the function.
• Set the second derivative equal to zero to find potential critical points.
• Solve for \( x \) in the resulting equation to list possible inflection points.
• Analyze intervals around each point to determine if there's a sign change, confirming an inflection point.

This structured method ensures that you can efficiently solve for areas where a function's concavity changes, providing deeper insights into the function's overall behavior.