Question

Use the intermediate value theorem to show that the polynomial function has a zero in the given interval: f(x)=10x^2 −22x−1; Interval: [1, 3]

Answer 1

To use the intermediate value theorem, evaluate the function at the endpoints of the interval and check for a sign change. If there is a sign change, a zero exists in the interval.

Step-by-step explanation:

To use the intermediate value theorem to show that the polynomial function f(x) = 10x^2 - 22x - 1 has a zero in the interval [1, 3], we need to find two points in the interval where the function takes opposite signs. First, evaluate the function at the endpoints of the interval:

• For x = 1, f(1) = 10(1)^2 - 22(1) - 1 = -13
• For x = 3, f(3) = 10(3)^2 - 22(3) - 1 = 4

Since the function takes opposite signs at x = 1 and x = 3, by the intermediate value theorem, there must exist at least one zero of the function in the interval [1, 3].