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 TheoremsThm. 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

2

1

-1

2 4

Every y-coordinate between the endpoints is hit

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 af c

b a

Here’s a demonstration.

Ex. Let f (x) = x2 + 2x – 1. Find c on the interval [-1,2] that satisfies MVT.