Problem 82

Question

Is it true that the concavity of the graph of a twice-differentiable function $y=f(x)$ changes every time $f^{\prime \prime}(x)=0 ?$ Give reasons for your answer.

Step-by-Step Solution

Verified

Answer

No, concavity changes only if $f''(x)$ changes signs at the point.

1Step 1: Understanding Concavity

The concavity of a function at a point tells us about the curve's behavior. A function is concave up where its second derivative, $f''(x)$, is greater than 0, and concave down where $f''(x)$ is less than 0. A change in concavity occurs at points called inflection points.

2Step 2: Investigating the Second Derivative Zeroes

To determine if the concavity changes when $f''(x) = 0$, check the behavior of $f''(x)$ around these points. If $f''(x)$ changes signs around these points, the function changes concavity there.

3Step 3: Possible Outcomes When $f''(x) = 0$

When $f''(x) = 0$, it doesn't necessarily mean the concavity changes. It must change sign, indicating an inflection point. Without a sign change, such as if $f''(x) < 0$ on both sides of the zero, the concavity remains the same.

4Step 4: Conclusion on the Given Statement

The statement that concavity always changes when $f''(x) = 0$ is false. Concavity changes only if $f''(x)$ changes signs; otherwise, the concavity stays the same despite the second derivative being zero.

Key Concepts

Second Derivative TestInflection PointsSign Change Analysis

Second Derivative Test

The second derivative test is a powerful tool used to determine concavity and points of inflection for a function. Simply put, the second derivative, denoted as $f''(x)$, gives us insight into how the slope of a function is changing. Here’s what you need to know:

If $f''(x) > 0$, the function is concave up at that point. Imagine it as being like a smiley face $:)$, curving upwards.
If $f''(x) < 0$, the function is concave down. This is much like a frown $:()$, curving downwards.
If $f''(x) = 0$, more investigation is needed. This might mean the function is transitioning between concave up and concave down or vice versa, signaling a potential inflection point.

In scenarios where $f''(x) = 0$, make sure to look at points nearby to see if concavity changes from positive to negative or negative to positive, as this determines an inflection point. Keep in mind, just because $f''(x) = 0$, it doesn't automatically mean a change in concavity occurs there.

Inflection Points

Inflection points are where a function changes its concavity. If a curve goes from being concave up to concave down, or vice versa, it’s an inflection point. These points are significant because they show where the function's curvature is shifting direction.To identify an inflection point:

First, find where $f''(x) = 0$.
Then, conduct a sign change analysis around those points to confirm a shift in concavity.
If there's a sign change in $f''(x)$ from positive to negative or from negative to positive as you move through the point, you've confirmed an inflection point.
If no sign change occurs, the point where $f''(x) = 0$ is not an inflection point.

Remember: not every point where the second derivative equals zero is an inflection point. The true test lies in the sign change.

Sign Change Analysis

Sign change analysis is a critical step when evaluating whether a change in concavity occurs at a given point. Essentially, it involves checking how the second derivative behaves right before and right after a point where $f''(x) = 0$. Here is a straightforward approach:

Choose a small interval around the zero point, say $x = a$, and evaluate $f''(x)$ for values slightly less than and greater than $a$.
If $f''(x)$ turns from positive to negative or negative to positive around $a$, this indicates a change in concavity, confirming it's an inflection point.
If $f''(x)$ does not change sign, the concavity remains consistent, implying no inflection point, even if $f''(x) = 0$.

This method is straightforward and intuitive, ensuring that students can convincingly identify real points of inflection through careful analysis of $f''(x)$ sign changes. So, always remember to check the intervals around pivotal points for any sign changes.

Problem 82

Problem 83

Other exercises in this chapter

Problem 81

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Problem 82

Solve the initial value problems in Exercises $67-86$. $$ \frac{d^{2} s}{d t^{2}}=\frac{3 t}{8} ;\left.\quad \frac{d s}{d t}\right|_{t=4}=3, \quad s(4)=4 $$

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Problem 83

Solve the initial value problems in Exercises $67-86$. $$ \frac{d^{3} y}{d x^{3}}=6 ; \quad y^{\prime \prime}(0)=-8, \quad y^{\prime}(0)=0, \quad y(0)=5 $$

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Problem 83

Quadratic curves What can you say about the inflection points of a quadratic curve $y=a x^{2}+b x+c, a \neq 0 ?$ Give reasons for your answer.

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