what is mean value therom

if a function f is continuous on the closed interval [a, b], where 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.

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Mean Value Theorem. Let f be a function which is differentiable on the closed interval [a, b]. Then there exists a point c in (a, b) such that Corollary. Let f be a differentiable function such that the derivative f `is positive on the closed interval [a, b]. Then f is increasing on [a, b]. Let f be a differentiable function such that the derivative f` is negative on the closed interval [a, b]. Then f is decreasing on [a, b]. Discussion [Using Flash] First Derivative Test. Suppose that c is a critical point of the function f and suppose that there is an interval (a, b) containing c. If f `(x) > 0 for all x in (a, c) and f`(x) < 0 for all x in (c, b), then c is a local maximum of f. If f `(x) < 0 for all x in (a, c) and f`(x) > 0 for all x in (c, b), then c is a local minimum of f.

Mean Value Theorem. Let f be a function which is differentiable on the closed interval [a, b]. Then there exists a point c in (a, b) such that Corollary. Let f be a differentiable function such that the derivative f `is positive on the closed interval [a, b]. Then f is increasing on [a, b]. Let f be a differentiable function such that the derivative f` is negative on the closed interval [a, b]. Then f is decreasing on [a, b]. Discussion [Using Flash] First Derivative Test. Suppose that c is a critical point of the function f and suppose that there is an interval (a, b) containing c. If f `(x) > 0 for all x in (a, c) and f`(x) < 0 for all x in (c, b), then c is a local maximum of f. If f `(x) < 0 for all x in (a, c) and f`(x) > 0 for all x in (c, b), then c is a local minimum of f.

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