### About The Intermediate Value Theorem:

The intermediate value theorem tells us that 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 the test is inconclusive.

a) f(x) = 2x^{4} + 10x^{3} + 14x^{2} + 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 the test is inconclusive.

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 the test is inconclusive.

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 the test is inconclusive.

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) inconclusive

This problem demonstrates a common source of confusion in this section. When we look at the Upper and Lower Bounds Theorem, it gives us a one-sided test. In other words, if a number passes the test and is shown to be an upper or lower bound, then we can conclude with certainty that it is an upper or lower bound. The reverse is not true. Desmos Link for More Detail Notice from the graph of our function shown above that -1 is a zero and a lower bound, however, it does not pass the lower bound test. Keep this in mind if you are using this strategy to find the zeros for a polynomial function. Again, if the number passes the test, then you can conclude that it is an upper or lower bound. If it fails, then it's really inconclusive. We would put the answer of inconclusive here since it does not pass the test for a lower bound even though it is actually a lower bound.

b) upper bound

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#4:

Solutions:

a) inconclusive

b) lower bound

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#5:

Solutions:

a) inconclusive

b) lower bound