Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x³−8x²+x+2; between 2 and 3

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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 \(f(c) = 0\).

Identify the interval given: between 2 and 3, so \(a = 2\) and \(b = 3\).

Evaluate the polynomial \(f(x) = 3x^3 - 8x^2 + x + 2\) at the endpoints: calculate \(f(2)\) and \(f(3)\).

Check the signs of \(f(2)\) and \(f(3)\). If one is positive and the other is negative, then by the IVT, there is at least one real zero between 2 and 3.

Conclude that since \(f\) is a polynomial (and thus continuous everywhere) and the function values at 2 and 3 have opposite signs, there must be a real zero of \(f(x)\) between 2 and 3.

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주요 개념

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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 polynomial's values at the given interval endpoints. If the function values have opposite signs, it indicates the function crosses zero, confirming the existence of a root between those points.

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x3−8x2+x+2; between 2 and 3