Polynomial functions and rates of change

A polynomial function is a sum of terms $a_n x^n + \dots + a_1 x + a_0$ with real coefficients and whole-number powers. The largest power with a nonzero coefficient is the degree; its term is the leading term. Polynomials are smooth and unbroken: no jumps, no asymptotes, no corners.

Two kinds of landmark matter in this topic. A local maximum or minimum is a point where the function switches between increasing and decreasing: a hilltop or valley floor compared to its immediate neighborhood. A global maximum or minimum is the largest or smallest value the function ever takes, anywhere.

If a polynomial has two different real zeros, the graph touches the axis at both. To get from one axis-touch to the other, the graph must rise (or dip) away from the axis and come back: somewhere strictly between the zeros there is at least one local extremum. This is a guarantee, not a tendency.

Be precise about what is not guaranteed: nothing forces another zero between them, nothing says the graph stays on one particular side of the axis, and the turning point between zeros is an extremum, not necessarily anything else. Zeros constrain turning points; they do not multiply themselves.

End behavior (fully treated in Topic 1.6) already earns one payoff here. An even-degree polynomial has both ends doing the same thing. Positive leading coefficient: both ends rise to $+\infty$, so somewhere in the middle the function bottoms out at a global minimum. Negative leading coefficient: both ends fall, so the function tops out at a global maximum.

Odd degree offers no such deal: one end rises and the other falls, so an odd-degree polynomial has no global maximum and no global minimum. Its local hilltops are only local; the function eventually climbs past every one of them. A local maximum of height 6 caps its neighborhood, never the whole cubic.

A point of inflection is where the graph changes concavity: the rate of change switches from increasing to decreasing, or the reverse. It is an event in the rate layer, not the value layer. The function need not turn around there, need not cross the axis there, and usually does neither.

1. Local extremum: the function switches direction (increasing ↔ decreasing).
2. Point of inflection: the rate switches trend (rate increasing ↔ rate decreasing); the bend flips.
3. Zero: the output is 0; the graph meets the axis.

Three different events, three different locations in general. A polynomial of degree $n$ has at most $n - 1$ local extrema and at most $n - 2$ points of inflection, and tables with equally spaced inputs still tell degree: the layer at which differences go constant is the degree, one more payoff of the differences habit from Topic 1.3.