Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x³−x−1; between 1 and 2

Verified step by step guidance

1. Recall the Intermediate Value Theorem (IVT), which states that if a function \( f \) is continuous on a closed interval \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one \( c \) in \((a, b)\) such that \( f(c) = 0 \).
2. Identify the function and the interval: \( f(x) = x^3 - x - 1 \), and the interval is \([1, 2]\).
3. Evaluate \( f(1) \): calculate \( f(1) = 1^3 - 1 - 1 = 1 - 1 - 1 \).
4. Evaluate \( f(2) \): calculate \( f(2) = 2^3 - 2 - 1 = 8 - 2 - 1 \).
5. Check the signs of \( f(1) \) and \( f(2) \). If one is negative and the other is positive, then by the IVT, there is at least one real zero of \( f(x) \) between 1 and 2.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes values f(a) and f(b) at each end, then it must take any value between f(a) and f(b) at some point within the interval. This theorem is used to prove the existence of roots within an interval.

Polynomial Continuity

Polynomials are continuous functions for all real numbers, meaning there are no breaks, jumps, or holes in their graphs. This continuity ensures that the Intermediate Value Theorem can be applied to polynomials on any interval.

Evaluating Function Values at Interval Endpoints

To apply the Intermediate Value Theorem, you calculate the function values at the endpoints of the interval. If the function values have opposite signs, it indicates the function crosses zero somewhere between those points, confirming the existence of a real root.