Problem 119 - University Calculus: Early Transcendentals

Question

Suppose that the second derivative of the function \(y=f(x)\) is $$y^{\prime \prime}=(x+1)(x-2).$$ For what \(x\) -values does the graph of \(f\) have an inflection point?

Step-by-Step Solution

Answer

The graph has inflection points at \(x = -1\) and \(x = 2\).

Step 1: Understand the Inflection Point

An inflection point occurs where the second derivative changes sign. This means we need to find where \(y'' = 0 \) and check the intervals around these points.

Step 2: Set the Second Derivative to Zero

We have \(y'' = (x+1)(x-2)\). To find the critical points, we set \(y'' = 0\). Therefore, \[(x+1)(x-2) = 0\]Solving this gives the potential inflection points \(x = -1\) and \(x = 2\).

Step 3: Test the Sign Change Around Critical Points

Check the sign of \(y''\) around the critical points to confirm the sign change, thus indicating inflection points:- For \(x < -1\), say \(x = -2\): \((x+1)(x-2) = (-2+1)(-2-2) = (-1)(-4) = 4\) (positive).- For \(-1 < x < 2\), say \(x = 0\): \((x+1)(x-2) = (0+1)(0-2) = (1)(-2) = -2\) (negative).- For \(x > 2\), say \(x = 3\): \((x+1)(x-2) = (3+1)(3-2) = (4)(1) = 4\) (positive).The sign changes around \(x = -1\) and \(x = 2\), confirming inflection points at these \(x\)-values.

Key Concepts

Second Derivative

The second derivative, denoted as \(y''\), is a mathematical tool used to understand how the rate of change of a function's slope changes. It provides information on whether a function is concave up, concave down, or undergoing a transition between these states, which occurs at inflection points. To find inflection points, we first need to determine where the second derivative equals zero. In the equation \(y'' = (x+1)(x-2)\), setting \(y'' = 0\) results in identifying potential x-values \(x = -1\) and \(x = 2\). These are the potential inflection points because here the rate of change could flip, changing the concavity of the graph.

Sign Change

Sign changes in the second derivative are crucial for determining the inflection points of a function. An inflection point exists where the second derivative changes signs, indicating a shift in concavity.To check for such changes, we examine the sign of \(y''\) at intervals around the critical points. By substituting values from each interval, such as \(x = -2\), \(x = 0\), and \(x = 3\), we find that the sign of \( (x+1)(x-2)\) changes from positive to negative to positive as \(x\) passes through \(-1\) and \(2\).• For \(x < -1\), \(y''\) is positive.• For \(-1 < x < 2\), \(y''\) is negative.• For \(x > 2\), \(y''\) is positive.These changes confirm that inflection points occur at \(x = -1\) and \(x = 2\).

Critical Points

In the context of second derivatives, critical points are x-values where the second derivative equals zero or is undefined. These are key locations where an inflection point might be present because it's where the direction of concavity can change.For the exercise at hand, setting the second derivative \(y'' = (x+1)(x-2) = 0\) produces the critical points \(x = -1\) and \(x = 2\). These values don't automatically ensure an inflection point, but they are where we start our examination of possible concavity changes.To confirm inflection points, it's essential to check the "sign change" around these critical points, as discussed earlier.

Function Analysis

Function analysis involves examining the behavior and properties of a function both through its algebraic formulation and graphical representation. When analyzing functions, derivatives play a significant role in understanding curvature and inflection points.The second derivative provides insights into the graph's concave nature:

• When \(y'' > 0\), the function is concave up.
• When \(y'' < 0\), the function is concave down.
• When \(y'' = 0\), a possible inflection point occurs.

For the exercise, we determined inflection points at \(x = -1\) and \(x = 2\). Checking these intervals for sign changes allows complete analysis and comprehension of where the function's curvature transitions, solidifying our understanding of the function's behavior.