Finding Absolute Extrema: Practice Problems & Overview

Instructor: Peter Kosek

Peter has taught Mathematics at the college level and has a master's degree in Mathematics.

In this lesson, you'll learn how to calculate absolute extrema, or the points that correspond to the largest and smallest value of a function on an interval. Read on to find out how the formula for absolute extrema can be used to solve equations related to mountain tops, space shuttles, and other real-life problems.

Absolute Extrema: A Real-Life Example

Let's suppose you want to take a ride on a space shuttle. The shuttle zooms into space. The Earth's gravity starts to slow it down but an extra set of turbo boosters kick in and propel you out into the galaxy. The velocity of your shuttle, in miles per hour, is modeled by the function, v(t) = t^3 - 75t^2 + 1800t, where t is in seconds. Can you determine the absolute extrema, or maximum and minimum velocities, between 10 and 32 seconds into your flight?

Let's first think about what it is that we're looking for. To find the maximum velocity, we're looking for a velocity, or a value of our function v(t), that is larger than any other velocity in this given time interval. Similarly, to find the minimum velocity, we're looking for a velocity that is smaller than any other velocity in this given time interval. In mathematical terms, we're looking for the absolute maximum and absolute minimum.

An absolute maximum of a function on an interval occurs at the point(s) at which the value of the function is greater than or equal to any other point in the interval. Similarly, absolute minimum of a function on an interval occurs at the point(s) at which the value of the function is less than or equal to any other point in the interval.

Calculating Absolute Extrema

In the space shuttle example above, we touched on some of the steps for determining the absolute maximum and minimum. To find the absolute extrema of a function, f, on a closed interval, [a, b], you'll need to complete the three calculations outlined below.

Step One: Find the critical points of f that are in the interval [a, b].

Step Two: Plug the critical point you found in step one, as well as the endpoints of our interval, a and b, into the function.

Step Three: Choose the value(s) of x that gave you the largest function value to be the absolute maximum(s) and the value(s) of x that gave you the smallest function value to be the absolute minimum(s).

Absolute Extrema on a Graph

What would this absolute maximum look like when you envision the function on a graph? How about a mountain top or the point in which all nearby values of the function are below it? The second possibility is that the absolute maximum occurs at either t = 10 or t = 32, such as the endpoints of our domain. If you're unsure about the second possibility, imagine a function that is a straight line going upward. If we choose an interval and asked for the absolute maximum on that interval, it would have to appear on one of the endpoints! There are no mountain tops since the function keeps going up and up!

Graphical Example of a Mountain Top

Mountain top example

Above is an example of a mountain top at the point x = 0.

Critical Points in Absolute Extrema

Now, how do we use calculus to determine what points could be our mountain tops? First, we'll find the critical points inside the interval. A critical point is a point in our domain in which the derivative either equals zero or does not exist. The critical points are the only possible points where a mountain top can exist. Let's find the derivative of v(t). We see that v'(t) = 3t^2 - 150t + 1800.

Clearly, the derivative exists for every point between 10 and 32, so we don't find any critical points satisfying that condition. We then have to check where our derivative is equal to zero. We'll solve 3t^2 - 150t + 1800 = 0. Factoring the left hand side, we get 3(t-30)(t-20) = 0. Therefore, the derivative equals zero when t = 20 and t = 30. So, our only critical points occur when t = 20 and t = 30.

The only other points necessary to consider are the endpoints. Therefore, our absolute maximum must exist at either t = 10, t = 20, t = 30, or t = 32. How do we decide which of these corresponds to the absolute maximum? Since the absolute maximum is the largest value of the function on the interval, let's plug each of them back into our original function and see which one gives us the largest value. We see v(10) = 11500, v(20) = 14000, v(30) = 13500, and v(32) = 13568. Therefore, the absolute maximum occurs at t = 20 with an absolute maximum velocity of 14,000 miles per hour.

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