Question

Suppose $f$ is a continuous function defined on a closed interval $[a, b]$. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for $f$ ? (b) What steps would you take to find those maximum and minimum values?

Name: Problem 2, Chapter 4: Student Solutions Manual for Stewart's / Single Variable Calculus: Early Transcendentals
Uploaded: 2019-06-26T16:12:16-08:00
Duration: 1 min 38 s
Channel: Alejandra Hernandez
Description: Suppose f is a continuous function defined on a closed interval [a, b]. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f ? (b) What steps would you take to find those maximum and minimum values?

   Suppose $f$ is a continuous function defined on a closed interval $[a, b]$.
(a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for $f$ ?
(b) What steps would you take to find those maximum and minimum values?

Student Solutions Manual for Stewart's / Single Variable Calculus: Early Transcendentals

James Stewart 6th Edition

Chapter 4, Problem 2

Instant Answer

Solved by Expert Alejandra Hernandez

06/26/2019

Step 1

The theorem you are looking for is the Extreme Value Theorem, which states that if $ f $ is continuous on a closed interval $[a, b]$, then $ f $ attains both an absolute maximum and an absolute minimum on that interval. Show more…

Show all steps

Thanks for your feedback!

Suppose $f$ is a continuous function defined on a closed interval $[a, b]$. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for $f$ ? (b) What steps would you take to find those maximum and minimum values?

Play audio

Feedback

Alejandra Hernandez and 90 other educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must attain both a maximum and a minimum value on that interval. This theorem is fundamental in real analysis and ensures that optimization problems on closed intervals have solutions.

Critical Points

Critical points are points in the domain of a function where the derivative is zero or undefined. These points are important in finding extreme values because local extrema can occur at these points, so they must be included when searching for absolute extrema.

Endpoint Evaluation

In addition to examining critical points, it is essential to evaluate the function at the endpoints of the closed interval. This is because the absolute maximum or minimum may occur at the boundaries of the interval rather than at points where the derivative is zero.

Please give Ace some feedback