What conditions must be to satisfied for the Mean Value Theorem to be valid?
f(x) is continuous in the interval [a, b] and differentiable in the interval (a, b)
The Mean Value Theorem states that for a given interval [a, b], if f(x) is continuous on the interval [a, b] and differentiable on the interval (a, b), then there exists at least one value c in the interval (a, b) such that:
f(b) – f(a) = f'(c)(b – a)
In order for the Mean Value Theorem to be valid, the following conditions must be satisfied:
1. Continuity: The function f(x) must be continuous on the closed interval [a, b].
2. Differentiability: The function f(x) must be differentiable on the open interval (a, b), meaning that the limit of the difference quotient of f(x) as x approaches any point c in (a, b) exists.
3. Finite Interval: The interval [a, b] must have finite length, meaning that b – a is a finite non-zero value.
If all of these conditions are satisfied, then there exists at least one point c in the interval (a, b) such that f'(c) equals the average rate of change of f(x) over the interval [a, b]. This point c is sometimes called the mean or average value of f(x) on the interval [a, b].
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