Understanding the Intermediate Value Theorem

Which of the following best states the Intermediate Value Theorem?

Which of the following functions is continuous on the interval $$[0, 2]$$ ?

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?

Olivia is designing a ramp whose height at a distance $$x$$ meters from the start is given by $$f(x) = x^3 - x - 2$$ meters. She wants to find out in which interval the ramp crosses ground level, i.e., where $$f(x) = 0$$ . Which interval contains the point where the ramp meets the ground?

If $$f(x)$$ is continuous on $$[a, b]$$ and $$f(a) < 0 < f(b)$$ , what does the Intermediate Value Theorem guarantee?

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

Let $$f(x)$$ be continuous on $$[2, 5]$$ with $$f(2) = 7$$ and $$f(5) = -1$$ . Which of the following values must $$f(x)$$ take for some $$x$$ in $$[2, 5]$$ ?

If $$f(x)$$ is continuous on $$[0, 3]$$ and $$f(0) = 2$$ , $$f(3) = 8$$ , which of the following values is NOT guaranteed to be taken by $$f(x)$$ for some $$x$$ in $$[0, 3]$$ ?

Which of the following tables justifies the use of the Intermediate Value Theorem to show that $$f(x) = 0$$ for some $$x$$ in $$[1, 3]$$ ?

Suppose $$f(x)$$ is continuous on $$[a, b]$$ and $$f(a) = 4$$ , $$f(b) = 9$$ . For which value of $$k$$ does the Intermediate Value Theorem guarantee that $$f(x) = k$$ for some $$x$$ in $$[a, b]$$ ?