Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. See Example 5. ƒ(x)=x^4-4x^3-x+3; 0.5 and 1

Verified step by step guidance

1. Identify the polynomial function: \( f(x) = x^4 - 4x^3 - x + 3 \).
2. Evaluate the function at the first given number: Calculate \( f(0.5) \).
3. Evaluate the function at the second given number: Calculate \( f(1) \).
4. Check the signs of \( f(0.5) \) and \( f(1) \). If they are opposite, it indicates a sign change.
5. Apply the Intermediate Value Theorem: Since \( f(x) \) is continuous and there is a sign change between \( f(0.5) \) and \( f(1) \), there must be at least one real zero between 0.5 and 1.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, then it must take on every value between those two points at least once. This theorem is crucial for proving the existence of roots in polynomial functions, as polynomials are continuous everywhere.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. They can have multiple real zeros, and their behavior can be analyzed using calculus and algebraic techniques to determine where they cross the x-axis.

Real Zeros

Real zeros of a function are the values of x for which the function evaluates to zero. Finding real zeros is essential in understanding the behavior of polynomial functions, as these points indicate where the graph intersects the x-axis, which is critical for applications in various fields such as physics and engineering.