Proving the Intermediate Value Theorem for Continuous Functions with Examples and Explanation

If f is continuous on [a,b] and k is a number between f(. a) and f(. b), then there exists at least one number c such that f(. c)=k

To prove the statement, we’ll use the Intermediate Value Theorem which states:

“Let f be a continuous function on a closed interval [a, b], and let k be a number between f(a) and f(b)

To prove the statement, we’ll use the Intermediate Value Theorem which states:

“Let f be a continuous function on a closed interval [a, b], and let k be a number between f(a) and f(b). Then there exists at least one number c in the interval (a, b) such that f(c) = k.”

Proof:

1. Assume that f is a continuous function on [a, b] and k is a number between f(a) and f(b).

2. Without loss of generality, assume that f(a) < f(b), so k lies between f(a) and f(b). 3. Define a new function g(x) = f(x) - k. 4. The function g(x) is continuous on [a, b]. 5. Since g(a) = f(a) - k < 0 and g(b) = f(b) - k > 0, by the Intermediate Value Theorem for continuous functions, there must exist at least one number c in the interval (a, b) such that g(c) = 0.

6. This means that f(c) – k = 0, or equivalently, f(c) = k.

7. Thus, we have shown that there exists at least one number c in the interval (a, b) such that f(c) = k.

Therefore, the statement is proven true.

The Intermediate Value Theorem is a fundamental result in analysis that guarantees the existence of solutions for equations of the form f(x) = k when f is a continuous function and k lies between the function values at the endpoints of the interval.

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