Question 1

Given the polynomial function $$f(x) = 5x^4$ - 9x^2$ - 19$$, use the Intermediate Value Theorem to determine whether there is a real zero between $$x = -2$$ and $$x = 1. $$Which of the following is correct?

Accepted Answer

Yes, because $$f(-2)$$ and $$f(1)$$ have opposite signs, indicating a zero exists between them.

Question 2

Which of the following best describes the Intermediate Value Theorem as it applies to polynomial functions?

Accepted Answer

If a polynomial function is continuous on $$[a, b]$$ and $$f(a)$$ and $$f(b)$$ have opposite signs, then there is at least one real zero between $$a$$ and $$b.$$

Question 3

Suppose $$f(x) = 5x^4$ - 9x^2$ - 19. $$Calculate $$f(-2)$$ and $$f(1) to $$justify your answer using the Intermediate Value Theorem.

Accepted Answer

$$f(-2) = 41$$, $$f(1) = -23$$

Question 4

Why does the Intermediate Value Theorem guarantee a real zero for $$f(x) = 5x^4$ - 9x^2$ - 19$$ between $$x = -2$$ and $$x = 1$$?

Accepted Answer

Because $$f(x) is $$continuous and $$f(-2) \u003e 0$$ while $$f(1) \u003c 0.$$

Question 5

Which of the following is NOT a requirement for applying the Intermediate Value Theorem to a function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$f(x)$$ must be differentiable on $$[a, b].$$

Question 6

If $$f(x) = 5x^4$ - 9x^2$ - 19$$ and you know $$f(-2) = 41$$ and $$f(1) = -23$$, what can you conclude?

Accepted Answer

There is at least one real zero between $$x = -2$$ and $$x = 1.$$

Question 7

Which of the following statements is true about the polynomial $$f(x) = 5x^4$ - 9x^2$ - 19$$?

Accepted Answer

It is continuous for all real values of $$x.$$

Question 8

A real-world application: Suppose $$f(x)$$ models the profit (in thousands of dollars) for a company, where $$x is $$the number of years since 2020. If $$f(-2) = 41$$ and $$f(1) = -23$$, what does the Intermediate Value Theorem suggest about the company's profit between 2018 and 2021?

Accepted Answer

The company must have broken even (profit = 0) at least once between 2018 and 2021.

Question 9

If $$f(x) is a $$polynomial and $$f(a) \u003e 0$$ while $$f(b) \u003c 0$$ for $$a \u003c b$$, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is at least one $$c in$$ $$(a, b)$$ such that $$f(c) = 0.$$

Question 10

Which of the following is a correct application of the Intermediate Value Theorem?

Accepted Answer

If $$f(2) = 5$$ and $$f(5) = -3$$ for a continuous function $$f(x)$$, then $$f(x)$$ must be zero for some $$x in$$ $$(2, 5).$$

Using the Intermediate Value Theorem to Find Real Zeros of a Polynomial

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