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Maximums and Minimums: Finding Relative Extrema

So, how many relative mins and maxs does the typical polynomial critter have?

Don't know?  When in doubt, draw pictures!

Let's draw some possible shapes of

f( x ) = x^4 + some x stuff

Remember, we use how many times he can cross (his degree) to guide us.

an x^4 guy with four real zeros and three relative extrema an x^4 guy with two real zeros and three relative extrema

an x^4 guy with two real zeros and one relative minimum
a plain x^4

Hmm...  It looks like an x^4 guy can have, at most, 3 relative extrema.

What about f( x ) = x^5 + some x stuff ?
(I'll let you do the drawing.)

It looks like an x^5 guy can have, at most, 4 relative extrema.

I smell a theorem brewing!  (Either that or it's because I didn't shower this morning.)

A polynomial of degree n can have,
 at most, n - 1 relative extrema.