In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x⁵−x³−1; 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: here, f(x) = $x^5-x^3-1$, and the interval is [1, 2].
3. Evaluate f at the endpoints: calculate f(1) and f(2) by substituting x = 1 and x = 2 into the function without simplifying the final numeric value.
4. Check the signs of f(1) and f(2): determine whether f(1) and f(2) are positive or negative to see if they have opposite signs.
5. Since f is a polynomial (which is continuous everywhere) and f(1) and f(2) have opposite signs, conclude by the Intermediate Value Theorem that 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.

Continuity of Polynomial Functions

Polynomial functions are continuous everywhere on the real number line, meaning there are no breaks, jumps, or holes in their graphs. This property 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's values at the given interval endpoints. If the function values have opposite signs, it indicates the function crosses zero within the interval, confirming the existence of a real root.