Question

Which of the following statements (if any) must be true because of the Intermediate Value Theorem(IVT).

A.f(x)= x+2/x has at least one root in the interval [1,3] by IVT.
B.f(x)= x/x-2 has at least one root in the interval [1,3] by IVT.
C.f(x)=x³ −8 has at least one root in the interval [1,3] by IVT.
D.f(x)=x³ has at least one root in the interval [1,3] by IVT.
E.None of the above.

Answer 1

The Intermediate Value Theorem states that if a function is continuous on a closed interval, then it takes on every value between the function values at the endpoints. Using this theorem, we can determine which statements are true based on the provided functions and intervals. Based on the Intermediate Value Theorem, statements A, C, and D are true, so the correct answer is E. None of the above.

Step-by-step explanation:

The Intermediate Value Theorem (IVT) states that if a function is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b) at least once.

Let's consider each statement:

A. The function f(x) = x+2/x is continuous on [1,3]. Since f(1) = 3 and f(3) = 11/3, and the values 3 and 11/3 are both in the range, the statement is true.

B. The function f(x) = x/x-2 is not continuous on [1,3] because it has a vertical asymptote at x = 2, meaning it's not defined at x = 2. Therefore, this statement is false.

C. The function f(x) = x³ - 8 is continuous on [1,3]. Since f(1) = -7 and f(3) = 19, and the values -7 and 19 are both in the range, the statement is true.

D. The function f(x) = x³ is continuous on [1,3]. Since f(1) = 1 and f(3) = 27, and the values 1 and 27 are both in the range, the statement is true.

Based on the Intermediate Value Theorem, statements A, C, and D are true. Therefore, the correct answer is E. None of the above.