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

*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.

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 3x

^{2}. The derivative (using the power rule) is the first degree polynomial, 6x.

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:**

- Find all extreme values for the entire range,
- Calculate the value of the polynomial at each of the extremes.
- 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.

## References

Maxima and Minima. Whitman University Calculus Online. Retrieved from https://www.whitman.edu/mathematics/calculus_online/section05.01.html on October 12, 2018.

**CITE THIS AS:**

**Stephanie Glen**. "Extreme Values of a Polynomial" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/extreme-values-of-a-polynomial/

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