Question 1

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

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?

Accepted Answer

If a function is continuous on $$[a, b]$$ and $$N is $$any number between $$f(a)$$ and $$f(b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f(c) = N.$$

Question 3

Evaluate $$f(-2)$$ for $$f(x) = 5x^4$ - 9x^2$ - 19.$$

Accepted Answer

Question 4

Evaluate $$f(1)$$ for $$f(x) = 5x^4$ - 9x^2$ - 19.$$

Accepted Answer

$$f(1) = -23$$

Question 5

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 $$f(-2) \u003e 0$$ and $$f(1) \u003c 0$$, so the function changes sign.

Question 6

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f(c) = 0.$$

Question 7

Which of the following is a necessary condition for the Intermediate Value Theorem to apply?

Accepted Answer

The function must be continuous on the interval.

Question 8

If $$f(x) is a $$polynomial, what can you say about its continuity?

Accepted Answer

It is continuous everywhere.

Question 9

A real-world application: The temperature $$T(t) in a $$chemical reaction is modeled by a continuous function. If $$T(2) = 50$$°C and $$T(5) = 100$$°C, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is a time $$t$$ between 2 and 5 where $$T(t) = 75$$°C.

Question 10

If $$f(x) = 5x^4$ - 9x^2$ - 19$$, which of the following intervals is guaranteed by the Intermediate Value Theorem to contain a real zero?

Accepted Answer

$$[-2, 1]$$

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

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