For f(x) = x^3 – 4x – 7, use the Intermediate Value Theorem to determine which interval must contain a zero of f.
Read my previous answer, given about three days ago. Last time, you (or someone else) gave four intervals and I told you which one had the zero.
My previous answer is at:
http://www.jiskha.com/display.cgi?id=1248395006
I described how to get the answer there.
To use the Intermediate Value Theorem, we need to understand what it states. The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on two values, f(a) and f(b), then it must also take on every value between f(a) and f(b). In simpler terms, if a function changes sign between two points on an interval, then it must have at least one zero on that interval.
Now let's apply the Intermediate Value Theorem to the given function f(x) = x^3 – 4x – 7.
Let's evaluate the function at the endpoints of different intervals to check for sign changes:
f(0) = (0)^3 – 4(0) – 7 = -7
f(1) = (1)^3 – 4(1) – 7 = -10
By evaluating the function at f(0) and f(1), we can see that there is a sign change between these two points. Since the function changes sign on the interval [0, 1], we can conclude that there must be at least one zero of the function f(x) = x^3 – 4x – 7 in that interval.
Therefore, the interval [0, 1] must contain a zero of f(x) = x^3 – 4x – 7.