Warm up Problems 1. Find and classify all critical points of f (x) = 4x 3 – 9x 2 – 12x + 3. 2. Find the absolute max./min. values of f (x) on the interval.

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Presentation transcript:

Warm up Problems 1. Find and classify all critical points of f (x) = 4x 3 – 9x 2 – 12x 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. Every y-coordinate between the endpoints is hit

Ex. Show that f (x) = x 5 – 3x 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 Here’s a demonstration.

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