# Mean-Value Theorem

## Rolle’s Theorem: Let a function be f: [a, b]®R, such that f is continuous over the interval [a, b], f is differentiable over the interval (a, b) and f(a) = f(b). Then there exists at least one real number, c in (a, b) such that f’(c) = 0. Geometrical Interpretation of Rolle’s Theorem: There is at least one point in [a, b] at which the tangent is parallel to x-axis. There may exist more than one point in the [a, b] at which the tangents are parallel to x-axis. Lagrange’s Mean Value Theorem: Let a function f: [a, b]®R, such that f  is continuous over the interval [a, b] and f  is differentiable over the interval (a, b). Then there exists at least one real number c in (a, b) such that f’(c) = [f(b) – f(a)]/( b- a). Lagrange’s mean value theorem asserts that there is at least one point lying between ‘x= a’ and ‘x= b’ at which the tangent is parallel to the chord AB where coordinates of A are (a, y¬1) and coordinates of B are (b, y1).The Mean Value Theorem is useful in finding the specific point of a continuous function that has the same slope a s the secant line. It is used to find the instantaneous rate and widely used to prove other theorems in calculus like Fundamental Theorem of Calculus.    Keywords:  Rolle’s Theorem, Geometrical Interpretation of Rolle’s theorem, Lagrange’s Mean Value Theorem

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State the Lagrange’s Mean Value Theorem.

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Theorem: If a function f(x) is continuous in the closed interval [a, b], derivable in the open interval (a, b), then there exists at least one real number c in (a, b) such that