First Derivative of the Function

Oct 13, 2025 by windailys.com 8 views

Introduction

In calculus, the concavity of a function is an important concept that helps us understand the behavior of the function. A function is said to be concave upward if its second derivative is positive, and concave downward if its second derivative is negative. An inflection point is a point on the graph of a function where the concavity changes. In this article, we will find the open intervals where the function $f(x) = 2x(x+5)^2$ is concave upward or concave downward, and find any inflection points for the function.

First Derivative of the Function

To find the concavity of the function, we first need to find its first derivative. The first derivative of the function $f(x) = 2x(x+5)^2$ can be found using the product rule and the chain rule.

f(x) = 2x(x+5)^2
f'(x) = 2(x+5)^2 + 2x(2(x+5))
f'(x) = 2(x+5)^2 + 4x(x+5)
f'(x) = 2(x+5)(x+5 + 2x)
f'(x) = 2(x+5)(3x+5)

Second Derivative of the Function

To find the concavity of the function, we need to find its second derivative. The second derivative of the function $f(x) = 2x(x+5)^2$ can be found by differentiating the first derivative.

f'(x) = 2(x+5)(3x+5)
f''(x) = 2(3x+5) + 2(x+5)(3)
f''(x) = 6x+10 + 6x+30
f''(x) = 12x+40

Concavity of the Function

Now that we have the second derivative of the function, we can find the concavity of the function. The function is concave upward if its second derivative is positive, and concave downward if its second derivative is negative.

f''(x) = 12x+40

To find the intervals where the function is concave upward or concave downward, we need to find the values of x for which the second derivative is positive or negative.

Solving the Inequality

To find the intervals where the function is concave upward or concave downward, we need to solve the inequality $f''(x) > 0$ and $f''(x) < 0$ .

f''(x) = 12x+40 > 0
12x+40 > 0
12x > -40
x > -40/12
x > -10/3
f''(x) = 12x+40 < 0
12x+40 < 0
12x < -40
x < -40/12
x < -10/3

However, we can see that the inequality $f''(x) > 0$ and $f''(x) < 0$ is the same. This is because the second derivative is a linear function, and the inequality is a linear inequality. Therefore, we can conclude that the function is concave upward for all values of x.

Inflection Points

An inflection point is a point on the graph of a function where the concavity changes. To find the inflection points of the function, we need to find the values of x for which the second derivative is equal to zero.

f''(x) = 12x+40 = 0
12x+40 = 0
12x = -40
x = -40/12
x = -10/3

Therefore, the inflection point of the function is $x = -10/3$ .

Conclusion

In this article, we found the open intervals where the function $f(x) = 2x(x+5)^2$ is concave upward or concave downward, and found any inflection points for the function. We found that the function is concave upward for all values of x, and the inflection point of the function is $x = -10/3$ .

Introduction

In our previous article, we discussed the concavity and inflection points of the function $f(x) = 2x(x+5)^2$ . In this article, we will answer some frequently asked questions about the concavity and inflection points of the function.

Q: What is concavity?

A: Concavity is a concept in calculus that describes the shape of a function. A function is said to be concave upward if its second derivative is positive, and concave downward if its second derivative is negative.

Q: What is an inflection point?

A: An inflection point is a point on the graph of a function where the concavity changes. In other words, it is a point where the function changes from being concave upward to concave downward, or vice versa.

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

A: To find the concavity of a function, you need to find its second derivative. If the second derivative is positive, the function is concave upward. If the second derivative is negative, the function is concave downward.

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 values of x for which the second derivative is equal to zero.

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

A: A concave upward function is a function that is curved upward, while a concave downward function is a function that is curved downward.

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. However, a function can have multiple inflection points, which means that it can change from being concave upward to concave downward, or vice versa, 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 use a graphing calculator or a computer program. You can also use a piece of graph paper and a pencil to draw the graph by hand.

Q: What is the significance of inflection points in real-world applications?

A: Inflection points are significant in real-world applications because they can help us understand the behavior of a function. For example, in economics, inflection points can help us understand the behavior of a company's profits or losses. In physics, inflection points can help us understand the behavior of a system's energy or momentum.

Q: Can a function have multiple inflection points?

A: Yes, a function can have multiple inflection points. This means that the function can change from being concave upward to concave downward, or vice versa, multiple times.

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

A: To find the inflection points of a function with multiple inflection points, you need to find the values of x for which the second derivative is equal to zero. You can do this by solving the equation $f''(x) = 0$ .

Conclusion

In this article, we answered some frequently asked questions about the concavity and inflection points of the function $f(x) = 2x(x+5)^2$ . We hope that this article has helped you understand the concept of concavity and inflection points, and how to apply them in real-world applications.