Finding the absolute extrema of a function is an important step when analyzing the behavior of a function.
Absolute extrema are the values a function reaches as it approaches either its highest or lowest value. These max and min values can be at any point of the function, whether it be at a point or along a line segment.
When analyzing functions, it is common to only look for absolute extrema on open intervals, but in this article, we will discuss how to find absolute extremoa on closed intervals as well.
When finding the absolute extremum of a function on a closed interval, there are two main ways to do this. The first way is to find all of the zeroes of the derivative of the function and then check whether those points are local max or min points. If they are, then they are absolute max or min values. The second way is to find all of the points where the derivative changes from negative to positive or positive to negative and check if those points are local max or min points.
Check each end point for extrema
Once you have the interval for the graph, you must check each end point of the interval for extrema. An extremum is a maximum or minimum value of the function on the interval.
If an extremum exists at the end point, then you can safely say the function is constant on the interval. That means there is no extremum on the interval, and it is not changing.
For example, if the function is y = 3×2/3 − 2x and the x-value equals 1, then y equals −2 which is not a constant value. There is an extremum at x = 1, so this cannot be said to be constant on [−1, 1].
Calculate the sign of the derivative at each point
The next step is to calculate the sign of the derivative at each point. The sign represents what type of value comes next: a minimum or maximum or a point where the function does not change at all.
If the derivative is negative, then the next value of the function is negative. If the derivative is positive, then the next value of the function is positive. If the derivative is zero, then there is no change in value at that point.
For example, look at the graph below. The red dot indicates where the graph crosses over from left to right (positive to negative) and where it crosses over from bottom to top (negative to positive).
The graph above shows several points where there are no changes in value, called stationary points. These are points where either the derivative is zero or there is no longer a direction change in the graph.
Compare the relative magnitude of the second derivative to the magnitude of x
Once you have the second derivative, you can determine the absolute extrema of the function. The second derivative represents the rate of change of the rate of change, or how fast the function is changing.
Absolute extrema occur when the function has a maximum or minimum value, when it reaches a point where it does not change in value any more. This occurs when the graph hits an horizontal line, vertical line, or gets to a point where it reverses direction.
To find absolute extrema, you need to compare the magnitude of the second derivative to the magnitude of x. If x is greater than any point on an horizontal line, then there is no absolute minimum; if it is less than any point on an horizontal line then there is no absolute maximum. If it equals a point on an horizontal line then there is an absolute minimum or maximum at that point.
Determine whether it is a local or global maximum or minimum
Once you have found the absolute value maximum or minimum on the closed interval, you can determine whether it is a local or global maximum or minimum.
A local maximum or minimum occurs when the function has a larger value on a neighboring interval. A global maximum or minimum occurs when the function has the largest value overall, no matter what interval you look at.
For example, if the function values increase on an open interval, then it is a global maximum. If the function values decrease on an open interval, then it is a global minimum.
Note that there can be local minima that are greater in value than global minima, and vice versa for maxima.
Confirm your results with calculus (this part is not necessary)
Once you have found the absolute extrema of your function, you can confirm your results with calculus.
If one of the values is a minimum or maximum, you can use the fundamental theorem of calculus to confirm that it is an extremum. If both are, then you can use the first derivative to show that it is an extremum.
For example, if x = 1 is an extremum and f (x) = 3×2/3 − 2x, then we can calculate the derivative and see that it equals zero at x = 1, which confirms that it is an extremum.
Another example would be y = 3×2/3 − 2x at x = 1 where y derivative would show that it equals zero at x = 1, confirming it as an absolute minimum.
Use technology to help with calculations (this part is not necessary)
Technology can be very helpful in finding the absolute extrema of functions. There are many apps that offer free trials or free downloads, and some even have a fair amount of features.
Some of the best app features include:
ability to input a function into the app,
ability to enter a closed interval and find the absolute extrema on that interval,
ability to enter a set of values and find the absolute minimum and maximum value on those values,
ability to graph the function and determine where minima and maxima are, and whether they are local or global.
Answer questions about your results (this part is not necessary)
Absolute extrema are found by looking at the graph and answering questions about the graph. For example, what are the lowest points on the graph? What are the highest points on the graph?
At what points is the slope zero? At what points is the slope negative one, or one point where it is positive and one where it is negative? What axis does the curve cross, above or below?
By looking at these questions you can find absolute extrema. Once you have found an absolute extremum, you can check if it is a maximum or minimum by asking another question: Is this point above or below the x-axis? If it is above the x-axis, it is a maximum, if it is below the x-axis, it is a minimum.
Checking whether a point is an extremum or not can be tricky. There are two ways to do this: One way is to find all of the absolute extrema and then check if any of them are not maxima or minima by checking if they are above or below the x-axis.
Share your post on social media (this part is not necessary)!
Absolute extrema of a function on a closed interval are found by finding the minimum and maximum values the function takes on the closed interval.
To do this, first find the derivative of the function and then set it to zero. You then solve for what the minimum or maximum value is by evaluating what value the derivative is equal to.
For example, let’s look at some examples:
If f(x) = x2 − x + 1, find the absolute extrema of f(x) on [1, 2].
Solution: The absolute maxima occur at x = 1 and x = 2, where f′(x) = 2 and 0, respectively. The absolute minimum occurs at x = −1, where f′(x) = −1. Thus 1
Leave a Reply