### About The Intermediate Value Theorem:

The intermediate value theorem tells us: if f(a) and f(b) are different signs, then there must be at least one real zero between a and b. For the upper and lower bounds, we can use synthetic division to test potential values and determine if they are an upper bound, meaning there won't be a real zero above or a lower bound, meaning there won't be a real zero below.

Test Objectives
• Demonstrate the ability to use the intermediate value theorem
• Demonstrate the ability to find an upper bound
• Demonstrate the ability to find a lower bound
Intermediate Value Theorem Practice Test:

#1:

Instructions: Show there is a zero between the two numbers given.

a) f(x) = x5 - 3x4 + 4x3 - 12x2 - 12x + 36
-1, 2

b) f(x) = x3 + 7x2 + 13x + 3
-4, 0

#2:

Instructions: Determine if k is an upper bound, a lower bound, or neither.

a) f(x) = 2x4 + 10x3 + 14x + 6x
k = 2

b) f(x) = x5 - x3 - 2x2 + 5x - 3
k = -5

#3:

Instructions: Determine if k is an upper bound, a lower bound, or neither.

a) f(x) = 4x5 - 16x4 - 8x3 + 16x2 + 4x
k = -1

b) f(x) = 2x4 - 2x3 - 5x2 - 3x + 2
k = 3

#4:

Instructions: Determine if k is an upper bound, a lower bound, or neither.

a) f(x) = x4 + 7x3 - 25x2 + 11x + 6
k = 1

b) f(x) = 4x5 - 11x4 - 18x3 - 7x2 - 4x
k = -4

#5:

Instructions: Determine if k is an upper bound, a lower bound, or neither.

a) f(x) = 3x4 - 14x3 + 11x2 - 14x + 8
k = 1

b) f(x) = 3x4 - x3 - 2x2 + 3x + 2
k = -4

Written Solutions:

#1:

Solutions:

a) f(-1) = 28, f(2) = -20, f(-1) > 0, f(2) < 0

b) f(-4) = -1, f(0) = 3, f(-4) < 0, f(0) > 0

#2:

Solutions:

a) upper bound

b) lower bound

#3:

Solutions:

a) neither

b) upper bound

#4:

Solutions:

a) neither

b) lower bound

#5:

Solutions:

a) neither

b) lower bound