Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. See Example 5. ƒ(x)=3x^2-x-4; 1 and 2

Verified step by step guidance

1. Identify the polynomial function: \( f(x) = 3x^2 - x - 4 \).
2. Evaluate the function at the first given number: Calculate \( f(1) \).
3. Evaluate the function at the second given number: Calculate \( f(2) \).
4. Check the signs of \( f(1) \) and \( f(2) \): If they are opposite, then by the Intermediate Value Theorem, there is at least one real zero between 1 and 2.
5. Conclude that since \( f(x) \) changes sign between \( x = 1 \) and \( x = 2 \), there is a real zero in the interval (1, 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 on different signs at the endpoints, then there exists at least one c in (a, b) such that f(c) = 0. This theorem is crucial for proving the existence of real zeros in polynomial functions.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. They are continuous and smooth, which means they do not have breaks or sharp turns, making them suitable for applying the Intermediate Value Theorem.

Evaluating Function Values

To apply the Intermediate Value Theorem, one must evaluate the polynomial function at the given endpoints of the interval. By calculating f(1) and f(2) for the function f(x) = 3x^2 - x - 4, we can determine if the function changes sign, indicating the presence of a real zero between these two points.