Question

Given the function f(x)= x⁴-6x³-x²-30x+4,use the Intermediate Value Theorem to decide which of the following intervals contains at least one zero. Select all that apply.

a) [−2,−1]
b) [−1,0]
c) [0,1]
d) [1,2]

Answer 1

To use the Intermediate Value Theorem, we evaluate f(x) for the given intervals and check if the function changes sign. The intervals that contain at least one zero are [-2, -1], [0, 1], and [1, 2].

Step-by-step explanation:

To use the Intermediate Value Theorem to decide which intervals contain at least one zero for the function f(x) = x⁴ - 6x³ - x² - 30x + 4, we need to evaluate f(x) for the given intervals and check if the function changes sign.

a) For the interval [-2, -1], f(-2) = -46 and f(-1) = 42. Since f(-2) is negative and f(-1) is positive, this interval contains at least one zero.

b) For the interval [-1, 0], f(-1) = 42 and f(0) = 4. Since f(-1) is positive and f(0) is positive, this interval does not contain a zero.

c) For the interval [0, 1], f(0) = 4 and f(1) = -32. Since f(0) is positive and f(1) is negative, this interval contains at least one zero.

d) For the interval [1, 2], f(1) = -32 and f(2) = 2. Since f(1) is negative and f(2) is positive, this interval contains at least one zero.

Therefore, the intervals that contain at least one zero are a) [-2, -1], c) [0, 1], and d) [1, 2].