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

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

Accepted Answer

$$f(1) = -23$$

Question 3

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 4

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) = 3$$, $$f(b) = -2. $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

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

Question 5

If $$f(x) is $$not continuous on $$[a, b]$$, can the Intermediate Value Theorem be applied?

Accepted Answer

No, continuity is required for the theorem to hold.

Question 6

A chemical reaction's rate is modeled by $$f(x) = 5x^4$ - 9x^2$ - 19. If $$the rate must be zero at some temperature between $$x = -2$$ and $$x = 1$$, what mathematical theorem justifies this?

Accepted Answer

Intermediate Value Theorem

Question 7

Which of the following is a necessary condition for the Intermediate Value Theorem to apply to a function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$f(x) is $$continuous on $$[a, b].$$

Question 8

If $$f(x) = 5x^4$ - 9x^2$ - 19$$ models the height (in meters) of a projectile at time $$x (in $$seconds), and you know $$f(-2) \u003e 0$$ and $$f(1) \u003c 0$$, what can you conclude?

Accepted Answer

The projectile reaches ground level at some time between $$x = -2$$ and $$x = 1.$$

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

Study Guide - Practice Questions