Polynomial Shape and Extrema

A line never turns. A parabola turns once, from decreasing to increasing or back. A degree 5 polynomial can turn up to 4 times, because its graph is built from more terms that each fight for control as x changes. Every time the function stops increasing and starts decreasing, it leaves behind a local maximum, the top of a hill. Every time it stops decreasing and starts increasing, it leaves a local minimum, the bottom of a valley. A degree n polynomial has at most n-1 turning points, though it can have fewer.

Turning points and zeros are linked. If a polynomial is negative at one real zero's neighborhood and positive at the next, it must climb from one to the other, and somewhere in between it has to stop climbing, at least if it later needs to come back down to hit a third zero. More precisely, between any two distinct real zeros where the function has the same behavior pattern, calculus guarantees at least one point where the slope is zero, a candidate for a local extremum. Take f(x) = x⁴ - 8x^2. Its derivative is 4x³ - 16x, zero at x = 0 and x = ±2. Check values: f(-2) = -16, f(0) = 0, f(2) = -16. So x = -2 and x = 2 are local minima and global minima at -16, while x = 0 is a local maximum at 0.

That last point matters: x = 0 is only a local maximum, not a global one, since the function heads off to positive infinity as x gets large in either direction. This is the hinge into end behavior and the leading term, which decides where the graph ultimately goes, no matter how many hills and valleys sit in the middle.