Calculus How To

Extreme Values of a Polynomial; Extreme Value Theorem

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Contents:

  1. Extreme Values of a Polynomial
  2. Extreme Value Theorem

Extreme Values of a Polynomial

Extreme values of a polynomial are the peaks and valleys of the polynomial—the points where direction changes.

polynomial extreme values

A third degree polynomial, with two extreme values: peak A and valley B.

The peaks and valleys are called relative (local) maxima and minima.

  • On a graph, you find extreme values by looking to see where there’s a mountain top (“peak”) or valley floor.
  • Mathematically, you find them by looking at the derivative. At an extreme point, where there is a direction change, the derivative of the function is zero.

Note: the derivative is the slope of the tangent line. In the above graph, the tangent line is horizontal, so it has a slope (derivative) of zero.

The Number of Extreme Values of a Polynomial

Polynomials can be classified by degree. This comes in handy when finding extreme values. A polynomial of degree n can have as many as n – 1 extreme values. For example, a 4th degree polynomial has 4 – 1 = 3 extremes.

The 4th degree polynomial (left ) has 3 extreme values; The second degree (right) has 1.




This follows directly from the fact that at an extremum, the derivative of the function is zero. If a polynomial is of n degrees, its derivative has n – 1 degrees. For example, take the 2nd degree polynomial 3x2. The derivative (using the power rule) is the first degree polynomial, 6x.
extreme value derivative

The derivative function (blue) crosses the x-axis where the original function (green) has a relative minimum.


The above image demonstrates an important result of the fundamental theorem of algebra: a polynomial of degree n has at most n roots. Roots (or zeros of a function) are where the function crosses the x-axis; for a derivative, these are the extrema of its parent polynomial.

Note that the polynomial of degree n doesn’t necessarily have n – 1 extreme values—that’s just the upper limit. The actual number of extreme values will always be n – a, where a is an odd number.

Absolute Extreme Values of Polynomials

The absolute extreme values (also known as the global extreme values) of a polynomial are the absolute maxima and minima of the polynomial. These are the points where the function takes its largest and smallest values, period.

An absolute extreme value is also a relative extreme value.

To find the absolute extreme values of a polynomial:


  1. Find all extreme values for the entire range,
  2. Calculate the value of the polynomial at each of the extremes.
  3. Find the value of the polynomial at the endpoints of the range.

The point at which the polynomial is largest is the absolute maximum value; the point at which our polynomial is smallest is the absolute minimum value.

You can also simply graph the polynomial and make a visual judgement. In the image below, the polynomial has a relative maxima at 2 and relative minima at 4 and -2. The relative minima at -2 is also a global minima; the absolute maxima doesn’t exist because the value of the polynomial goes toward positive infinity at both ends.

polynomial third degree


Extreme Value Theorem

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.

Extreme Value Theorem
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.

Extreme Values of a Polynomial: References

Wandzura, Jacqueline and Wandzura, Stephen. Extreme Value Theorem Demonstration. Retrieved from http://demonstrations.wolfram.com/ExtremeValueTheorem/ on August 11, 2019
Maxima and Minima. Whitman University Calculus Online. Retrieved from https://www.whitman.edu/mathematics/calculus_online/section05.01.html on October 12, 2018.

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