Now, for the relative minimums... Those are the bottoms of the valleys:
Relative mins are the lowest points in their little neighborhoods.
f has a relative min of -3 at x = -1.
f has a relative min of -1 at x = 4.
Find the relative extrema:
So, how many relative mins and maxes does the typical polynomial critter have?
Don't know? When in doubt, draw pictures!
Let's draw some possible shapes of
Remember, we use how many real zeros he might have to guide us.
Hmm... It looks like an guy can have, at most, 3 relative extrema.
(I'll let you do the drawing.)
It looks like an 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.