1. Rolle’s Theorem: Let f be a function that satisfies the following 3 hypotheses: 1.f is continuous on the closed interval [a,b]. 2.f is differentiable on the open interval (a,b). 3.f(a)=f(b) Then there is a number c in (a,b) such that f ‘ (c)=0. The Mean Value Theorem: Let f be a function that satisfies the following hypotheses: 1.f is continuous on the closed interval [a,b]. 2.f is differentiable on the open interval (a,b). Then there is a number c in (a,b) such that 1. or, equivalently, 2.. Theorem: If f ‘ (x)=0 for all x in an interval (a,b), then f is constant on (a,b). Corrolary: If f ‘ (x) = g ‘ (x) for all x in an interval (a,b), then f—g is constant on (a,b); that is f(x)=g(x)+c wher c is a constant. Calculus Notes 4.2: The Mean Value Theorem.

2. Example 1: Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. Calculus Notes 4.2: The Mean Value Theorem. 1. Since f is a polynomial, f is continuous on all, and so is continuous on [0,2] 2. Since f is a polynomial, f is differentiable on all, and so is differentiable (0,2) Both are in (0,2)

3. Example 2: Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conlcusion of the Mean Value Theorem. Calculus Notes 4.2: The Mean Value Theorem. 1. Since f is a polynomial, f is continuous on all, and so is continuous on [-1,1] 2. Since f is a polynomial, f is differentiable on all, and so is differentiable (-1,1) c=0, which is in (-1,1).

4. Example 3: Show that the following equation has at most two real roots. Calculus Notes 4.2: The Mean Value Theorem. Suppose that f(x)=x 4 +4x+c has three distinct real roots a, b, d where a<b<d. Then f(a)=f(b)=f(d)=0. Has as its only real solution x=-1. Thus, f(x) can have at most two real roots. By Rolle’s Theorem there are numbers c 1 and c 2 with and and, so must have at least two real solutions. However PS 4.2 pg.238 #1, 7, 11, 12, 16, 22, 23, 32, 33(try) (8)