### 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

#1:

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

a) f(x) = x^{5} - 3x^{4} + 4x^{3} - 12x^{2} - 12x + 36

-1, 2

b) f(x) = x^{3} + 7x^{2} + 13x + 3

-4, 0

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#2:

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

a) f(x) = 2x^{4} + 10x^{3} + 14x + 6x

k = 2

b) f(x) = x^{5} - x^{3} - 2x^{2} + 5x - 3

k = -5

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#3:

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

a) f(x) = 4x^{5} - 16x^{4} - 8x^{3} + 16x^{2} + 4x

k = -1

b) f(x) = 2x^{4} - 2x^{3} - 5x^{2} - 3x + 2

k = 3

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#4:

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

a) f(x) = x^{4} + 7x^{3} - 25x^{2} + 11x + 6

k = 1

b) f(x) = 4x^{5} - 11x^{4} - 18x^{3} - 7x^{2} - 4x

k = -4

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#5:

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

a) f(x) = 3x^{4} - 14x^{3} + 11x^{2} - 14x + 8

k = 1

b) f(x) = 3x^{4 }- x^{3} - 2x^{2} + 3x + 2

k = -4

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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

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#2:

Solutions:

a) upper bound

b) lower bound

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#3:

Solutions:

a) neither

b) upper bound

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#4:

Solutions:

a) neither

b) lower bound

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#5:

Solutions:

a) neither

b) lower bound