Minima, Maxima & Points of Inflection (College Board AP® Precalculus): Study Guide

Local minima and maxima

What are local minima and maxima?

• A polynomial function has a local (or relative) maximum at a point where the function switches from increasing to decreasing

  • The output value at that point is greater than the output values at all nearby points

  • On the graph, a local maximum appears as the top of a "hill"

• A polynomial function has a local (or relative) minimum at a point where the function switches from decreasing to increasing

  • The output value at that point is less than the output values at all nearby points

  • On the graph, a local minimum appears as the bottom of a "valley"

• Local maxima and minima are collectively called local extrema

• A local extremum also occurs at an included endpoint of a polynomial with a restricted domain

  • I.e. if the domain is restricted to a closed interval , then the function values at or will be local extrema

What are global minima and maxima?

• Global minima and maxima are defined as follows:

  • A global (or absolute) maximum is a local maximum whose output value is greater than every other output value of the function

  • A global (or absolute) minimum is a local minimum whose output value is less than every other output value of the function

• Note that not every polynomial function has a global maximum or global minimum

  • E.g. a cubic function (or other odd degree polynomial) extends to in one direction and in the other, so it has no global maximum or minimum

• However, polynomial functions of even degree will always have either a global maximum or a global minimum

  • If the leading coefficient is positive (parabola opens upward, or similar shape for higher even degrees), the function has a global minimum

  • If the leading coefficient is negative (parabola opens downward, or similar shape for higher even degrees), the function has a global maximum

What is the relationship between zeros and local extrema?

• Between every two distinct real zeros of a nonconstant polynomial function

  • there must be at least one input value corresponding to a local maximum or local minimum

• This makes sense graphically

  • If the graph crosses (or touches) the -axis at two different points, it must turn around at least once between them

• This is a useful fact for sketching and reasoning about polynomial graphs

  • E.g. if a polynomial has zeros at and

  • then there must be at least one local maximum or local minimum for some value of between and

Points of inflection

What is a point of inflection?

• A polynomial function has a point of inflection at an input value where

  • the rate of change of the function

  • changes from increasing to decreasing or from decreasing to increasing

• In terms of the function's graph, this is where the graph changes from concave up to concave down, or from concave down to concave up

  • Recall from the Concavity study guide that

    • Concave up means the rate of change is increasing (the graph curves upward)

    • Concave down means the rate of change is decreasing (the graph curves downward)

• At a point of inflection, the direction of the curvature changes

  • The graph changes from "holding water" to "spilling water", or vice versa

How can I identify a point of inflection on a graph?

• Look for where the curvature of the graph changes direction

  • The graph changes from bending one way to bending the other way

• A point of inflection is not a maximum or minimum

  • The function does not need to switch between increasing and decreasing at a point of inflection

  • The function can be increasing (or decreasing) on both sides of a point of inflection

• What changes is how quickly it is increasing or decreasing

  • I.e. whether the rate of change is speeding up or slowing down

How do points of inflection relate to local extrema?

• Points of inflection and local extrema describe different features of a polynomial's graph

  • Local extrema occur where the function switches between increasing and decreasing

  • Points of inflection occur where the rate of change of the function switches between increasing and decreasing (i.e. where concavity changes)

• For higher-degree polynomials, points of inflection often occur between local maxima and local minima

  • E.g. for a cubic polynomial with one local maximum and one local minimum, there is a point of inflection between them where the concavity changes