Section 5.5The Intermediate Value Theorem

Rolle’s TheoremThe Mean Value Theorem

3.6

Intermediate Value Theorem (IVT)

If f is continuous on [a, b] and N is a value between f(a) and f(b), then there is at least one point c between a and b where f takes on the value N.

a

f(a)

b

f(b)N

c

Rolle’s Theorem

If f is continuous on [a, b], if f(a) = 0, f(b) = 0, then there is at least one number c on (a, b) where f ‘ (c ) = 0

a b

slope = 0

c

f ‘ (c ) = 0

Given the curve:

x

f(x)

x+h

f(x+h)

sec

f x h f xm

h

tan

h 0

f x h f xm lim

h

The Mean Value Theorem (MVT)aka the ‘crooked’ Rolle’s Theorem

If f is continuous on [a, b] and differentiable on (a, b)There is at least one number c on (a, b) at which

f b f af ' c

b a

ab

f(a)

f(b)

c

Conclusion:Slope of Secant Line

EqualsSlope of Tangent Line

2 f b f aIf f x x 2x 1, a 0, b 1, and f ' c , find c.

b a

f(0) = -1 f(1) = 2

f b f a 2 13

b a 1 0

f ' x 2x 2

3 2x 2 1

x2

Find the value(s) of c which satisfy Rolle’s Theorem for on the interval [0, 1]. 4f x x x

Verify…..f(0) = 0 – 0 = 0 f(1) = 1 – 1 = 0

3f ' x 4x 1 30 4x 1

31

c4

which is on [0, 1]

Find the value(s) of c that satisfy the Mean Value Theorem for

1f x x on 4, 4

x

1 17 1 17f 4 4 f 4 4

4 4 4 4

17 17f b f a 174 4

b a 4 4 16

2

17 1f ' c 1

16 x

Find the value(s) of c that satisfy the Mean Value Theorem for

1f x x on 4, 4

x

Note: The Mean Value Theorem requires the function to be continuous on [-4, 4] and differentiable on (-4, 4). Therefore, sincef(x) is discontinuous at x = 0 which is on [-4, 4], there may be no value of c which satisfies the Mean Value Theorem

Since has no real solution, there is no value of c on

[-4, 4] which satisfies the Mean Value Theorem

2

1 1

16 x

Given the graph of f(x) below, use the graph of f to estimate thenumbers on [0, 3.5] which satisfy the conclusion of the Mean ValueTheorem.

2Determine whether f x x 2x 2 satisfies the hypothesis of

the Mean Value Theorem on -2, 2 . If it does, find all numbers

f b f ac in (a, b) such that f ' c

b a

f(x) is continuous and differentiable on [-2, 2]

f 2 f 2 6 2

2 2 42

f ' x 2x 2

2x 2 2 c 0

On the interval [-2, 2], c = 0 satisfies the conclusion of MVT

2x 1Determine whether f x satisfies the hypothesis of

x 2the Mean Value Theorem on -2, 1 . If it does, find all numbers

f b f ac in (a, b) such that f ' c

b a

f(x) is continuous and differentiable on [-2, 1]

30

f 1 f 2 41 2 3

1

4

2

2

2x x 2 1 x 1

xf ' x

2

2

2

x 4x 1 1

x 4x 4 4

2 24x 16x 4 x 4x 4 23x 12x 0

3x x 4 0 On the interval [-2, 1], c = 0 satisfies the conclusion of MVT

2x 1Determine whether f x satisfies the hypothesis of

x 2the Mean Value Theorem on 0, 4 . If it does, find all numbers

f b f ac in (a, b) such that f ' c

b a

Since f(x) is discontinuous at x = 2, which is part of the interval[0, 4], the Mean Value Theorem does not apply

3Determine whether f x x 3x 1 satisfies the hypothesis of

the Mean Value Theorem on -1, 2 . If it does, find all numbers

f b f ac in (a, b) such that f ' c

b a

f(x) is continuous and differentiable on [-1, 2]

f0

2 f 1 3 3

2 1 3

23x 3f x' 23x 3 0

c = 1 satisfies the conclusion of MVT

3 x 1 x 1 0

f(3) = 39 f(-2) = 64

f b f a 64 395

b a 2 3

For how many value(s) of c is f ‘ (c ) = -5?

If , how many numbers on [-2, 3] satisfythe conclusion of the Mean Value Theorem.

2 2f x x 12 x 4

A. 0 B. 1 C. 2 D. 3 E. 4

CALCULATOR REQUIRED

X X X