Suppose Aiden is hiking up a mountain trail that starts at an elevation of $$-2$$ meters at the beginning ( $$x = 1$$ km) and reaches $$5$$ meters at the end ( $$x = 4$$ km). The elevation function $$f(x)$$ is continuous along the trail. According to the Intermediate Value Theorem, which of the following must be true?

$$f(x) = 3$$ for some $$x$$ in $$[1, 4]$$

$$f(x) = 0$$ for some $$x$$ in $$[1, 4]$$

$$f(x) = -3$$ for some $$x$$ in $$[1, 4]$$

$$f(x) = 6$$ for some $$x$$ in $$[1, 4]$$

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

$$f(x) = 0$$ for some $$x$$ in $$[a, b]$$

$$f(x) = 0$$ for all $$x$$ in $$[a, b]$$

$$f(x) > 0$$ for all $$x$$ in $$[a, b]$$

Which of the following statements is true about the function $$f(x) = \frac{1}{x-2}$$ on the interval $$[1, 3]$$ ?

$$f(x)$$ is not continuous on $$[1, 3]$$

$$f(x)$$ is continuous on $$[1, 3]$$

$$f(x)$$ is continuous at $$x = 2$$

$$f(x)$$ is defined for all $$x$$ in $$[1, 3]$$

Given $$f(x)$$ is continuous on $$[a, b]$$ and $$f(a) = f(b)$$ , which of the following is always true?

$$f(x)$$ takes every value between $$f(a)$$ and $$f(b)$$

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

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

$$f(x)$$ has a root in $$[a, b]$$

Which of the following is an example where the Intermediate Value Theorem cannot be applied?

$$f(x)$$ is not continuous on $$[0, 1]$$ , $$f(0) = 1$$ , $$f(1) = -1$$

$$f(x)$$ is continuous on $$[0, 1]$$ , $$f(0) = 1$$ , $$f(1) = -1$$

$$f(x)$$ is continuous on $$[0, 1]$$ , $$f(0) = 1$$ , $$f(1) = 1$$

$$f(x)$$ is continuous on $$[0, 1]$$ , $$f(0) = -1$$ , $$f(1) = 1$$

If $$f(x)$$ is continuous on $$[a, b]$$ and $$f(a) = 2$$ , $$f(b) = 6$$ , which of the following statements is true?

$$f(x)$$ must take the value $$4$$ for some $$x$$ in $$[a, b]$$

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

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

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

Resource Library

Math

Calculus

Intermediate Value Theorem

Understanding The Intermediate Value Theorem

Understanding the Intermediate Value Theorem

Authored by Alvaro Leal

Mathematics

11th Grade

CCSS covered

Used 1+ times

Understanding the Intermediate Value Theorem

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MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Which of the following best states the Intermediate Value Theorem?

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Which of the following is NOT a requirement for applying the Intermediate Value Theorem?

The function must be continuous on the interval.

The function must be differentiable on the interval.

The interval must be closed.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

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