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state lagrange mean value theorem and give its geometrical interpretation.is the therorem applicable to f(x)=|x-3|+2 in [2,3] |
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Answer» Answer: The Mean Value Theorem (MVT) Lagrange’s mean value theorem (MVT) states that if a function f ( x ) is continuous on a closed INTERVAL [ a , b ] and differentiable on the open interval ( a , b ) , then there is at least one point x = c on this interval, such that f ( b ) − f ( a ) = f ′ ( c ) ( b − a ) . This theorem (ALSO known as First Mean Value Theorem) allows to express the increment of a function on an interval through the value of the derivative at an intermediate point of the segment. Explanation: Consider the auxiliary function F ( x ) = f ( x ) + λ x . We choose a NUMBER λ such that the condition F ( a ) = F ( b ) is SATISFIED. Then f ( a ) + λ a = f ( b ) + λ b , ⇒ f ( b ) − f ( a ) = λ ( a − b ) , ⇒ λ = − f ( b ) − f ( a ) b − a . As a result, we have F ( x ) = f ( x ) − f ( b ) − f ( a ) b − a x . The function F ( x ) is continuous on the closed interval [ a , b ] , differentiable on the open interval ( a , b ) and takes equal values at the endpoints of the interval. Therefore, it satisfies all the conditions of Rolle’s theorem. Then there is a point c in the interval ( a , b ) such that F ′ ( c ) = 0. It follows that f ′ ( c ) − f ( b ) − f ( a ) b − a = 0 or f ( b ) − f ( a ) = f ′ ( c ) ( b − a ) . |
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