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