For f(x) = x^3 – 4x – 7, use the Intermediate Value Theorem to determine which interval

must contain a zero of f.

A. Between 0 and 1
B. Between 1 and 2
C. Between 2 and 3
D. Between 3 and 4

I am leaning towards Choice A. What does everyone think? I would appreciate some feed back! Thanks!

Calculate f(x) at 0,1,2,3 and 4 and see where it changes sign. That will be the answer. There must be a zero of f(x) in that interval, since it is an intermediate value.

By the way, your choice is wrong. f(x) is negative in interval A.

Ok I am trying to figure this out. If A is wrong then my next choice was

C. Between 2 and 3 Is this correct?

Assuming C is correct, could you explain based on what you made this choice?

To determine which interval must contain a zero of the function f(x) = x^3 – 4x – 7 using the Intermediate Value Theorem, we can evaluate the function at the endpoints of each interval and check the sign changes.

Let's check the values of f(x) at the endpoints of each interval:

For interval [0, 1]:
f(0) = (0)^3 – 4(0) – 7 = -7
f(1) = (1)^3 – 4(1) – 7 = -10

Since f(0) < 0 and f(1) < 0, there is no sign change in this interval, and it cannot contain a zero.

For interval [1, 2]:
f(1) = (1)^3 – 4(1) – 7 = -10
f(2) = (2)^3 – 4(2) – 7 = 1

Here, f(1) < 0 and f(2) > 0, indicating that there is a sign change within this interval. Consequently, by the Intermediate Value Theorem, there must be a zero of the function f(x) between 1 and 2.

Therefore, the correct option is B. Between 1 and 2.