1. Calculus Notes 4.2: The Mean Value Theorem.
Rolle’s Theorem: Let f be a function that satisfies the following 3 hypotheses:
f is continuous on the closed interval [a,b].
f is differentiable on the open interval (a,b).
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:
Then there is a number c in (a,b) such that 1.
or, equivalently, .
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.

2. Calculus Notes 4.2: The Mean Value Theorem.
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.
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. Calculus Notes 4.2: The Mean Value Theorem.
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.
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. Calculus Notes 4.2: The Mean Value Theorem.
Example 3: Show that the following equation has at most two real roots.
Suppose that f(x)=x4+4x+c has three distinct real roots a, b, d where a<b<d.
Then f(a)=f(b)=f(d)=0.
By Rolle’s Theorem there are numbers c1 and c2 with and and , so must have at least two real solutions.
However
Has as its only real solution x=-1. Thus, f(x) can have at most two real roots.
PS 4.2 pg.238 #1, 7, 11, 12, 16, 22, 23, 32, 33(try) (8)