Warm up Problems
1. Find and classify all critical points of
f (x) = 4x3 – 9x2 – 12x + 3.
2. Find the absolute max./min. values of
f (x) on the interval [-1,4].
Miscellaneous Theorems
Thm. Extreme Value Theorem
If f (x) is continuous on a closed interval, then
it has an absolute max. and an absolute min. on
the interval.
Thm. Intermediate Value Theorem
Let f (x) be continuous on the interval [a,b]. If
k is any number between f (a) and f (b), then
there is a point c on [a,b] such that f (c) = k.
3
Every y-coordinate
between the
endpoints is hit
2
1
2
-1
4
Ex. Show that f (x) = x5 – 3x2 + 1 has a
zero on the interval [-1,2].
Why did the chicken cross the road?
[Assume the chicken’s path is a continuous function
with starting point on one side of the road and ending
point on the other side of the road.]
Thm. Mean Value Theorem
If f (x) is continuous and differentiable on the
interval [a,b], then there is some point c on the
interval such that
f b   f  a 
f c 
ba
Here’s a demonstration.
Ex. Let f (x) = x2 + 2x – 1. Find c on the
interval [-1,2] that satisfies MVT.