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.

'

f b f af c

b a

'f b f a f c b a

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.

3 23 2 5, [0,2]f x x x x

1. Since f is a polynomial, f is continuous on all , and so is continuous on [0,2]

3. 0 5 2f f

2' 3 6 2f x x x

2. Since f is a polynomial, f is differentiable on all , and so is differentiable (0,2)

20 3 6 2x x

6 36 4 3 2

2 3x

6 12

6x

11 3

3x

Both are in (0,2)

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.

23 2 5, [ 1,1]f x x x

1. Since f is a polynomial, f is continuous on all , and so is continuous on [-1,1]

'

f b f af c

b a

' 6 2f x x

2. Since f is a polynomial, f is differentiable on all , and so is differentiable (-1,1)

10 6

'1 1

f c

' 2f c

' 2 6 2f c x 6 0

0

c

c

c=0, which is in (-1,1).

Example 3: Show that the following equation has at most two real roots.

Calculus Notes 4.2: The Mean Value Theorem.

4 4 0x x c Suppose that f(x)=x4+4x+c has three distinct real roots a, b, d where a<b<d.

3 3 20 ' 4 4 4 1 4 1 1f x x x x x x

1 20 ' 'f c f c

Then f(a)=f(b)=f(d)=0.

1a c b 2b c d ' 0f x

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 c1 and c2 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)