The degree of a polynomial will tell you its number of relative extrema and whether it has any absolute extrema.

Odd Degree Polynomial

Will have, at most, n – 1 relative extrema
No absolute extrema

Even Degree Polynomial

Will have, at most, n – 1 relative extrema
Has 0 or an even number of absolute extrema

You need to see the graph of the polynomial in order to determine the location of the extrema. You can sketch the graphs of polynomial functions by hand, or you can use a graphing utility such as your hand-held graphing calculator or online sites like Desmos.

Give it a shot!

In the activity below, practice using a graphing utility to find the coordinates of the extrema. Identify each of the extrema as a relative maximum, a relative minimum, an absolute maximum, or an absolute minimum. Click on the polynomial to check your answer.

\( y = -6x^3 + x^2 + 30x - 5 \)

GRAPH

EXTREMA

Relative Maximum
(1.348, 22.56)

Relative Minimum
(-1.237, -29.223)

\( y = x(x + 2)(x – 1) \)

GRAPH

EXTREMA

Relative Maximum
(-1.215, 2.113)

Relative Minimum
(0.549, -0.631)

\( y = -x^4-3x^3+4x^2 + 12x \)

GRAPH

EXTREMA

Relative Maximum
(-2.574, 2.879)

Relative Minimum
(-0.93, -6.035)

Absolute Maximum
(1.254, 12.949)

\( y = x^3-x \)

GRAPH

EXTREMA

Relative Maximum
(-0.577, 0.385)

Relative Minimum
(0.577, -0.385)

\( y = x^4+x^3-13x^2-x+12 \)

GRAPH

EXTREMA

Absolute Minimum
(-2.935, -48.128)

Relative Maximum
(0, 12)

Relative Minimum
(2.224, -19.059)