Extreme Value Theorem: Simple Definition

Calculus Definitions >

The extreme value theorem tells us that a continuous function contains both the maximum value and a minimum value as long as the function is:

1. Real-valued,
2. Defined on a closed interval, I.

Another way of saying this is that the continuous, real-valued function, f, attains its maximum value and its minimum value each at least once on the interval.

This theorem may seem too simple to be really important or significant, but it actually is the foundation for other theorems and is very significant in the groundwork of Calculus. It is used to prove Rolle’s theorem, among other things.

Examples of the Extreme Value Theorem in Action

The image below shows a continuous function f(x) on a closed interval from a to b. [a,b]. The absolute maximum is shown in red and the absolute minimum is in blue.

Since we know the function f(x) = x² is continuous and real valued on the closed interval [0,1] we know that it will attain both a maximum and a minimum on this interval.

In order for the extreme value theorem to be able to work, you do need to make sure that a function satisfies the requirements:

• Real-valued,
• Continuous,
• Closed interval domain,

For example, the function f(x) = 1/x on the half open interval (0,1] doesn’t attain a maximum. That’s because the interval is not closed.

History

The extreme value theorem itself was first proved by the Bohemian mathematician and philosopher Bernard Bolzano (of Bolzano Theorem’s fame) in 1830, but his book, Function Theory, was only published a hundred years later in 1930. Another mathematician, Weierstrass, also discovered a proof of the theorem in 1860.

References

Wandzura, Jacqueline and Wandzura, Stephen. Extreme Value Theorem Demonstration. Retrieved from http://demonstrations.wolfram.com/ExtremeValueTheorem/ on August 11, 2019

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