First Derivative and Second Derivative

Oct 13, 2025 by windailys.com 136 views

Introduction

In calculus, the concavity of a function refers to the shape of its graph. A function is said to be concave upward if its graph lies above its tangent line at every point, and concave downward if its graph lies below its tangent line at every point. An inflection point is a point on the graph where the concavity changes. In this article, we will find the open intervals where the function ${ f(x) = 15x - 15e^{-x} }$ is concave upward or concave downward, and also find any inflection points for the function.

First Derivative and Second Derivative

To determine the concavity of a function, we need to find its first and second derivatives. The first derivative of a function represents the slope of its tangent line at every point, while the second derivative represents the rate of change of the slope.

Let's find the first and second derivatives of the function ${ f(x) = 15x - 15e^{-x} }$ .

First Derivative

To find the first derivative of the function, we will use the power rule and the chain rule.

${ f'(x) = \frac{d}{dx} (15x - 15e^{-x}) }$

Using the power rule, we get:

${ f'(x) = 15 - 15(-e^{-x}) }$

Simplifying, we get:

${ f'(x) = 15 + 15e^{-x} }$

Second Derivative

To find the second derivative of the function, we will differentiate the first derivative.

${ f''(x) = \frac{d}{dx} (15 + 15e^{-x}) }$

Using the chain rule, we get:

${ f''(x) = 0 - 15(-e^{-x}) }$

Simplifying, we get:

${ f''(x) = 15e^{-x} }$

Concavity

A function is concave upward if its second derivative is positive, and concave downward if its second derivative is negative.

Let's find the intervals where the function is concave upward or concave downward.

Concave Upward Intervals

To find the intervals where the function is concave upward, we need to find the intervals where the second derivative is positive.

${ f''(x) = 15e^{-x} > 0 }$

Since ${ e^{-x} > 0 }$ for all ${ x }$ , we have:

${ 15e^{-x} > 0 }$

Therefore, the function is concave upward for all ${ x }$ .

Concave Downward Intervals

To find the intervals where the function is concave downward, we need to find the intervals where the second derivative is negative.

${ f''(x) = 15e^{-x} < 0 }$

Since ${ e^{-x} > 0 }$ for all ${ x }$ , we have:

${ 15e^{-x} < 0 }$

This is not possible, since the product of a positive number and a positive number is always positive. Therefore, the function is not concave downward for any ${ x }$ .

Inflection Points

An inflection point is a point on the graph where the concavity changes. To find the inflection points, we need to find the points where the second derivative is equal to zero.

${ f''(x) = 15e^{-x} = 0 }$

Since ${ e^{-x} > 0 }$ for all ${ x }$ , we have:

${ 15e^{-x} \neq 0 }$

Therefore, there are no inflection points for the function.

Conclusion

In this article, we found the open intervals where the function ${ f(x) = 15x - 15e^{-x} }$ is concave upward or concave downward, and also found any inflection points for the function. We found that the function is concave upward for all ${ x }$ , and there are no inflection points for the function.

References

Calculus by Michael Spivak
Calculus by James Stewart

Note

Introduction

In our previous article, we discussed the concavity and inflection points of a function. In this article, we will answer some frequently asked questions related to this topic.

Q: What is concavity?

A: Concavity refers to the shape of a function's graph. A function is said to be concave upward if its graph lies above its tangent line at every point, and concave downward if its graph lies below its tangent line at every point.

Q: How do I determine the concavity of a function?

A: To determine the concavity of a function, you need to find its first and second derivatives. The second derivative represents the rate of change of the slope of the function.

Q: What is an inflection point?

A: An inflection point is a point on the graph where the concavity changes. To find the inflection points, you need to find the points where the second derivative is equal to zero.

Q: How do I find the inflection points of a function?

A: To find the inflection points of a function, you need to find the points where the second derivative is equal to zero. You can do this by setting the second derivative equal to zero and solving for x.

Q: What is the difference between a concave upward and a concave downward function?

A: A concave upward function has a positive second derivative, while a concave downward function has a negative second derivative.

Q: Can a function be both concave upward and concave downward at the same time?

A: No, a function cannot be both concave upward and concave downward at the same time. A function is either concave upward or concave downward, but not both.

Q: How do I determine the concavity of a function at a specific point?

A: To determine the concavity of a function at a specific point, you need to find the second derivative of the function at that point. If the second derivative is positive, the function is concave upward at that point. If the second derivative is negative, the function is concave downward at that point.

Q: Can a function have multiple inflection points?

A: Yes, a function can have multiple inflection points. This occurs when the second derivative changes sign multiple times.

Q: How do I graph a function that is concave upward or concave downward?

A: To graph a function that is concave upward or concave downward, you need to find the first and second derivatives of the function. Then, you can use the second derivative to determine the concavity of the function at different points.

Conclusion

In this article, we answered some frequently asked questions related to the concavity and inflection points of a function. We hope this article has been helpful in understanding this topic.

References

Calculus by Michael Spivak
Calculus by James Stewart

Note

This article is for educational purposes only and is not intended to be a comprehensive treatment of the subject.