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Inflection Point: Definition & Examples

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will learn about the inflection points and concavity of a function. Through definition and example, we will solidify our understanding of these concepts.

Concavity

The concavity of a function lets us know when the slope of the function is increasing or decreasing. For instance, if we were driving down the road, the slope of the function representing our distance with respect to time would be our speed. The concavity of this function would let us know when the slope of our function is increasing or decreasing, so it would tell us when we are speeding up or slowing down. The concavity of a function refers to when the function's graph is curving upward or downward.

When a function is curving upward, its slope is increasing, and we say the function is concave up.

Concave Up
concave up

When a function is curving downward, its slope is decreasing, and we say the function is concave down.

Concave Down
concave down

The Second Derivative Test

We can determine whether a function is concave up or concave down using the second derivative test. The second derivative test says that when a function's second derivative is positive, the function is concave up, and when a function's second derivative is negative, the function is concave down. For example, consider the function f(x) = -2x^2. The first derivative of this function is f ' (x) = -4x, and the second derivative of this function is f ' ' (x) = -4. We see that the second derivative is negative, so the function is concave down.

Let's consider the function f(x) = 4x^3 + 3x^2 - 7x shown in the graph below.

Concavity
inflection point

We see that this function is concave up in some places and concave down in others. The first derivative of f is f ' (x) = 12x^2 + 6x - 7, and the second derivative of the function is f ' ' (x) = 24x + 6. Notice, the second derivative is positive for x > -1/4 and negative for x < -1/4. Therefore, the function f is concave up for x > -1/4 and concave down for x < -1/4, which can be observed in our graph.

Inflection Point

The inflection point of a function is where the function changes from concave up to concave down or vice versa. When we think about our driving example, the inflection points of the function representing our distance with respect to time would indicate when we start to slow down or when we start to speed up.

Inflection Point
inflection point

Finding the inflection points of a function involves first finding points that may be an inflection point, and then testing those points to determine which ones are inflection points. To find inflection points, we follow these steps.

1.) Find the second derivative of the function.

2.) Identify any points that make the second derivative equal to zero by setting the second derivative equal to zero and solving.

3.) Identify any points that make the second derivative undefined, such as a zero denominator or a negative under a square root.

4.) Take the points you found in steps 2 and 3 and find the sign of the second derivative on both sides of the point. If the second derivative is positive on one side and negative on the other, this indicates that the function would be concave up on one side of the point and concave down on the other side, showing that the point is an inflection point.

Let's look at an example to solidify our understanding of these steps.

Example

We saw the inflection point labeled in the graph of g(x) = x^3 earlier. Let's take this function through our steps to verify that this is an inflection point.

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