4 min read•february 15, 2024
Welcome back to Calculus with Fiveable! Let’s get into it.
Can the derivative of a function give us more information than just where the function increases or decreases? Yes, it can! In this study guide, we’ll learn how you can use the derivative of a function to determine its relative or local extrema.
We can use the First Derivative Test to determine the relative (local) extrema of a function. It states that if the derivative of a function changes from positive to negative at a point (which means that the function changes from increasing to decreasing at the point), then the function has a local or relative maximum at that point. If the derivative changes from negative to positive at a point (which means that the function changes from decreasing to increasing at the point), then the function has a local or relative minimum at that point.
The process of applying the First Derivative Test is similar to the process of finding the intervals where the function increases or decreases.
First, we need to find the critical points of the function. A critical point is a point where the derivative of the function is equal to 0 or is not defined. Once these points have been identified, we need to find the sign of the derivative of the function on either side of the critical point.
Let’s walk through an example together.
Consider the function . The derivative of this function is using the power rule. At , the derivative is equal to , which means that it is a critical point, and thus we should evaluate the function’s derivative on either side of the point.
To check the behavior of the function on the left, substitute a value less than 0 into . Let’s use
Now we know that the derivative is negative for
The process is similar here! Substitute a value greater than 0 into . Let’s plug in
The derivative is positive for .
The derivative of the function is negative on the left side of the point and the derivative of the function is positive on the right side, so the point is a local minimum. ⬇️
Take a look at the function and its local minimum!
Now it’s time for you to do some practice on your own!
Let be a polynomial function and let its derivative, , be defined as . Where are ’s relative minima?
Let . Where are ’s relative maxima?
Since the problem asks us to find ’s relative minima, we should apply the First Derivative Test. To do so, we first find the function’s critical points by setting equal to
By solving each of these, we know that the critical points of are and . This means that the function can have minima at only these points.
Then we need to evaluate the left and right sides of the function’s derivative at each of these points to determine which are the relative minima. Here’s a table summarizing the results of plugging in the appropriate values!
Left | Right | Verdict | |
Relative maximum | |||
Not an extremum | |||
Relative minimum |
In conclusion, has one relative minimum at
Since the problem asks us to find ’s relative maxima, we should apply the First Derivative Test. To do so, we first need to find the function’s critical points. Since , the critical points are and in this problem as are at these points. This means that the function can have maxima at only these points.
We then need to evaluate the left and right sides of the function’s derivative at each of these points to determine which are the relative maxima.
Left | Right | Verdict | |
Relative maximum | |||
Relative minimum |
In conclusion, has one relative maximum at
Great work! The Derivative Test is a great tool that allows us to determine relative extrema of a function by identifying critical points and analyzing the sign of the derivative on either side of those points. It's important to keep in mind the limitations of the test and use other techniques to determine global extrema.
Decreasing Function
: A decreasing function is a function in which the values of the dependent variable decrease as the values of the independent variable increase. In other words, as you move from left to right on the graph, the y-values go down.Derivative
: A derivative represents the rate at which a function is changing at any given point. It measures how sensitive one quantity is to small changes in another quantity.First Derivative Test
: The First Derivative Test is a method used to determine the intervals on which a function is increasing or decreasing, and to identify local extrema (maximum or minimum) points.Global Extremum
: A global extremum refers to the highest (global maximum) or lowest (global minimum) point on the entire graph of a function. These points represent the absolute maximum or minimum values of the function.Increasing Function
: An increasing function is a function in which the values of the dependent variable increase as the values of the independent variable increase. In other words, as you move from left to right on the graph, the y-values go up.Relative Maximum
: A relative maximum is a point on a graph where the function reaches its highest value within a specific interval. It is higher than all nearby points but may not be the absolute highest point on the entire graph.Relative Minimum
: A relative minimum is a point on a graph where the function reaches its lowest value within a specific interval. It is lower than all nearby points but may not be the absolute lowest point on the entire graph.Second Derivative Test
: The second derivative test is used to determine whether critical points correspond to local maxima, minima, or neither. It involves analyzing the concavity of a function at those critical points.4 min read•february 15, 2024
Welcome back to Calculus with Fiveable! Let’s get into it.
Can the derivative of a function give us more information than just where the function increases or decreases? Yes, it can! In this study guide, we’ll learn how you can use the derivative of a function to determine its relative or local extrema.
We can use the First Derivative Test to determine the relative (local) extrema of a function. It states that if the derivative of a function changes from positive to negative at a point (which means that the function changes from increasing to decreasing at the point), then the function has a local or relative maximum at that point. If the derivative changes from negative to positive at a point (which means that the function changes from decreasing to increasing at the point), then the function has a local or relative minimum at that point.
The process of applying the First Derivative Test is similar to the process of finding the intervals where the function increases or decreases.
First, we need to find the critical points of the function. A critical point is a point where the derivative of the function is equal to 0 or is not defined. Once these points have been identified, we need to find the sign of the derivative of the function on either side of the critical point.
Let’s walk through an example together.
Consider the function . The derivative of this function is using the power rule. At , the derivative is equal to , which means that it is a critical point, and thus we should evaluate the function’s derivative on either side of the point.
To check the behavior of the function on the left, substitute a value less than 0 into . Let’s use
Now we know that the derivative is negative for
The process is similar here! Substitute a value greater than 0 into . Let’s plug in
The derivative is positive for .
The derivative of the function is negative on the left side of the point and the derivative of the function is positive on the right side, so the point is a local minimum. ⬇️
Take a look at the function and its local minimum!
Now it’s time for you to do some practice on your own!
Let be a polynomial function and let its derivative, , be defined as . Where are ’s relative minima?
Let . Where are ’s relative maxima?
Since the problem asks us to find ’s relative minima, we should apply the First Derivative Test. To do so, we first find the function’s critical points by setting equal to
By solving each of these, we know that the critical points of are and . This means that the function can have minima at only these points.
Then we need to evaluate the left and right sides of the function’s derivative at each of these points to determine which are the relative minima. Here’s a table summarizing the results of plugging in the appropriate values!
Left | Right | Verdict | |
Relative maximum | |||
Not an extremum | |||
Relative minimum |
In conclusion, has one relative minimum at
Since the problem asks us to find ’s relative maxima, we should apply the First Derivative Test. To do so, we first need to find the function’s critical points. Since , the critical points are and in this problem as are at these points. This means that the function can have maxima at only these points.
We then need to evaluate the left and right sides of the function’s derivative at each of these points to determine which are the relative maxima.
Left | Right | Verdict | |
Relative maximum | |||
Relative minimum |
In conclusion, has one relative maximum at
Great work! The Derivative Test is a great tool that allows us to determine relative extrema of a function by identifying critical points and analyzing the sign of the derivative on either side of those points. It's important to keep in mind the limitations of the test and use other techniques to determine global extrema.
Decreasing Function
: A decreasing function is a function in which the values of the dependent variable decrease as the values of the independent variable increase. In other words, as you move from left to right on the graph, the y-values go down.Derivative
: A derivative represents the rate at which a function is changing at any given point. It measures how sensitive one quantity is to small changes in another quantity.First Derivative Test
: The First Derivative Test is a method used to determine the intervals on which a function is increasing or decreasing, and to identify local extrema (maximum or minimum) points.Global Extremum
: A global extremum refers to the highest (global maximum) or lowest (global minimum) point on the entire graph of a function. These points represent the absolute maximum or minimum values of the function.Increasing Function
: An increasing function is a function in which the values of the dependent variable increase as the values of the independent variable increase. In other words, as you move from left to right on the graph, the y-values go up.Relative Maximum
: A relative maximum is a point on a graph where the function reaches its highest value within a specific interval. It is higher than all nearby points but may not be the absolute highest point on the entire graph.Relative Minimum
: A relative minimum is a point on a graph where the function reaches its lowest value within a specific interval. It is lower than all nearby points but may not be the absolute lowest point on the entire graph.Second Derivative Test
: The second derivative test is used to determine whether critical points correspond to local maxima, minima, or neither. It involves analyzing the concavity of a function at those critical points.© 2024 Fiveable Inc. All rights reserved.
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