Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=x⁴-4x³-x+3; 0.5 and 1

Verified step by step guidance

1. Recall the Intermediate Value Theorem (IVT), which states that if a function \( f(x) \) 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 \( f(x) = x^4 - 4x^3 - x + 3 \) and the interval \([0.5, 1]\). Since \( f(x) \) is a polynomial, it is continuous everywhere, including on this interval.
3. Evaluate \( f(0.5) \) by substituting \( x = 0.5 \) into the function: \( f(0.5) = (0.5)^4 - 4(0.5)^3 - 0.5 + 3 \).
4. Evaluate \( f(1) \) by substituting \( x = 1 \) into the function: \( f(1) = (1)^4 - 4(1)^3 - 1 + 3 \).
5. Check the signs of \( f(0.5) \) and \( f(1) \). If one is positive and the other is negative, then by the IVT, there is at least one real zero of \( f(x) \) between 0.5 and 1.

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 when the function changes sign.

Polynomial Continuity

Polynomial functions are continuous everywhere on the real number line, meaning there are no breaks, jumps, or holes in their graphs. This continuity ensures that the Intermediate Value Theorem can be applied to any interval when analyzing polynomial functions.

Evaluating Function Values at Interval Endpoints

To apply the Intermediate Value Theorem, you must 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.