The Mean Value Theorem In Calculus: Explained With Conditions And Applications

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 (MVT) is a theorem in calculus that states that for a given continuous and differentiable function f(x) defined on an interval [a, b], there exists a point c in the interval (a, b) such that the slope of the tangent at c is equal to the average rate of change of the function over the interval [a, b].

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]. This means that the function must be able to be drawn without lifting the pen from the paper, and there must be no gaps, jumps or breaks in the graph.

2. Differentiability: The function f(x) must be differentiable on the open interval (a, b). This means that the limit of the difference quotient that defines the derivative exists at every point in the interval and is finite.

3. Non-zero interval length: The interval [a, b] must have a non-zero length. This means that the endpoints a and b cannot be the same.

When these conditions are satisfied, the mean value theorem guarantees the existence of a point c in the interval (a, b) such that f'(c) = (f(b)−f(a))/(b−a), or in other words, the average rate of change of the function over the interval is equal to the instantaneous rate of change at some point in the interval.

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