Question

Using the Intermediate Value Theorem (IVT), determine the property of the function f(x) = x⁴ - x³ + 4 on the interval [1,2] that guarantees the existence of a point where f(x) = 0.

Answer 1

The Intermediate Value Theorem (IVT) guarantees that the function f(x) = x⁴ - x³ + 4 will have a point where f(x) = 0 on the interval [1,2] if f(x) changes sign between f(1) and f(2).

Step-by-step explanation:

The Intermediate Value Theorem (IVT) states that if a function is continuous on a closed interval [a, b] and takes on two different values, f(a) and f(b), then it must also take on every value in between those two values at least once.

In the case of the function f(x) = x⁴ - x³ + 4 on the interval [1,2], if f(x) changes sign between f(1) and f(2), then by the IVT, there exists a point on the interval where f(x) = 0.

Let's calculate f(1) and f(2):
f(1) = 1⁴ - 1³ + 4 = 1 - 1 + 4 = 4
f(2) = 2⁴ - 2³ + 4 = 16 - 8 + 4 = 12

Since f(1) is positive and f(2) is positive, and the function is continuous, we can conclude that f(x) must equal 0 at some point on the interval [1,2].