Mean Value Theorem
1/(b-a) int(a,b) (f(x))
The Mean Value Theorem (MVT) is a fundamental theorem of calculus that states that for any continuous and differentiable function f(x) defined on an interval [a, b], where a and b are any two real numbers and a < b, there exists at least one point c between a and b, such that the slope of the tangent at point c is equal to the average slope of the function over the interval [a, b]. Mathematically, the Mean Value Theorem can be stated as follows: if f(x) is continuous on [a, b] and differentiable on (a, b), then there exists at least one c such that: f'(c) = (f(b) - f(a))/(b - a) The Mean Value Theorem illustrates the relationship between the average rate of change of a function and its instantaneous rate of change at any point within a certain range. One important application of the Mean Value Theorem is to find the maximum or minimum values of a function on an interval [a, b]. If we want to find the largest or smallest value of a continuous function f(x) on the interval [a, b], we can use the Mean Value Theorem to show that the maximum or minimum occurs at either endpoint or at a critical point within the interval. Overall, the Mean Value Theorem is a powerful tool in calculus that has various applications in optimization problems, curve sketching, and physics.
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