(MVT) Mean Value Theorem
f is continuous on [a,b]differentiable on (a,b)x=a and x=b joined by secant linethere must exist at least one point c between them such that the tangent line at c has the same slope as the secant linef'(c) = f(b)-f(a) / b-a
The Mean Value Theorem (MVT) is a theorem in calculus that relates the derivative of a function to its average or mean value. It states that for any differentiable function f(x) on a closed interval [a, b], there exists at least one point c in the interval such that the slope of the tangent at c is equal to the average slope of the function over [a, b]. In other words, if the function f(x) is continuous on the interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that:
f'(c) = (f(b) – f(a))/(b – a)
This theorem is important because it provides a way of linking the geometric and algebraic properties of a function. It guarantees the existence of at least one point where the derivative of a function is equal to the average rate of change of the function over an interval. It has many applications in calculus, such as in optimization problems, where it can be used to find extreme values and in proving other theorems like Rolle’s theorem.
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