Use the Intermediate Value Theorem to show that there exists a number c in the given interval su

step 1 Hello. In the equation in the function FX it is given FX is equal to X to M minus X plus 1 and t

Use the Intermediate Value Theorem to show that there exists...

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states that if a function is continuous on a closed interval [a, b] and N is any number between f(a) and f(b), then there exists at least one c in [a, b] such that f(c) = N. This theorem is fundamental in calculus for proving the existence of roots or other function values within an interval without necessarily computing the exact value initially.

Continuity of Polynomial Functions

Polynomial functions, such as f(x) = x² - x + 1, are continuous everywhere on the real number line. This continuity on any closed interval, including [-1, 4], ensures that the Intermediate Value Theorem can be applied. The inherent smoothness of polynomials guarantees that there are no breaks, jumps, or discontinuities where the theorem might fail.

Solving Quadratic Equations

To find the specific value c for which f(c) = M, the equation is set equal to M, resulting in a quadratic equation. Solving quadratic equations, typically by factoring, completing the square, or using the quadratic formula, allows us to determine the potential candidates for c. In this context, setting f(x) = 7 and solving the resultant quadratic is a direct application of this method.

Domain Considerations

After solving the quadratic equation derived from f(c) = M, it is important to check which solutions lie within the specified interval. Although a quadratic might yield two solutions, only the solution that falls within the given domain of the problem is valid for the application of the Intermediate Value Theorem. This verification step is crucial to ensure consistency with the problem’s conditions.

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