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 that there exists at least one number c such that f(c) = k, we can use the Intermediate Value Theorem (IVT)
To prove that there exists at least one number c such that f(c) = k, we can use the Intermediate Value Theorem (IVT).
The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a, b] and k is any 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.
Here’s a step-by-step proof:
1. Given that f is continuous on [a, b] and k is a number between f(a) and f(b).
2. We want to show that there exists at least one number c such that f(c) = k.
Proof by contradiction:
Assume that there exists no number c in [a, b] such that f(c) = k.
Since f(a) < k < f(b), then by the definition of the function values, f(a) - k < 0 and f(b) - k > 0.
Consider the function g(x) = f(x) – k.
g(a) = f(a) – k < 0
g(b) = f(b) - k > 0
Since f is continuous on [a, b], g(x) = f(x) – k is also continuous on [a, b].
Since g(a) < 0 and g(b) > 0, we can apply the Intermediate Value Theorem to the function g(x) on [a, b]. This means that there exists a number c in (a, b) such that g(c) = 0.
Now, consider g(c) = f(c) – k.
Since g(c) = 0, f(c) – k = 0, which implies that f(c) = k.
This contradicts our assumption that no number c exists in [a, b] such that f(c) = k.
Therefore, our assumption was false, and there exists at least one number c in [a, b] such that f(c) = k.
Thus, we have proven that 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.
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