Calculus How To

Extreme Values of a Polynomial

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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


Maxima and Minima. Whitman University Calculus Online. Retrieved from on October 12, 2018.

Stephanie Glen. "Extreme Values of a Polynomial" From Calculus for the rest of us!

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