For f (x) = x4 – 2x2 – 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
Calculate the values of f(x) at each pair of numbers. The answer will be the pair for which the value of f(x) changes sign.
f(0) = -7
f(1) = -8
f(2) = 16 - 8 -7 = +1
f(3) = 81 - 18 -7 = +56
f(4) = an even bigger positive number
etc.
Clearly the answer is C.
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).
To determine which interval must contain a zero of f(x) = x^4 - 2x^2 - 7, we can evaluate the function at the endpoints of each interval provided and check if the function changes sign between these values. If it does, then by the Intermediate Value Theorem, the function must have a zero in that interval.
Let's evaluate the function at the endpoints of each interval:
For interval A (between 0 and 1):
f(0) = (0)^4 - 2(0)^2 - 7 = -7
f(1) = (1)^4 - 2(1)^2 - 7 = -8
The function does not change sign between f(0) and f(1), and both values are negative. Therefore, we can conclude that there is no zero in this interval.
For interval B (between 1 and 2):
f(1) = -8
f(2) = (2)^4 - 2(2)^2 - 7 = 7
The function changes sign between f(1) and f(2), with f(1) being negative and f(2) being positive. Therefore, by the Intermediate Value Theorem, there must be a zero in this interval.
For interval C (between 2 and 3):
f(2) = 7
f(3) = (3)^4 - 2(3)^2 - 7 = 28
The function does not change sign between f(2) and f(3), and both values are positive. Therefore, there is no zero in this interval.
For interval D (between 3 and 4):
f(3) = 28
f(4) = (4)^4 - 2(4)^2 - 7 = 105
The function changes sign between f(3) and f(4), with f(3) being positive and f(4) being positive. Therefore, by the Intermediate Value Theorem, there must be a zero in this interval.
Based on the evaluations and the Intermediate Value Theorem, the interval that must contain a zero of f(x) is between 1 and 2, as option B suggests.