At an end point is greater or smaller than f c rolles

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at an end point is greater (or smaller) than f ( c ). Rolle’s Theorem. If f : [ a, b ] R is continuous and it is differentiable at every x ( a, b ) and f ( a ) = f ( b ) = 0 then there exists a point c in ( a, b ) at which f 0 ( c ) = 0. Proof. We know that a continuous function on a bounded closed interval attains its maximum and minimum values. If both these values are zero, the function is identically equal to zero and f 0 = 0 everywhere. If one of these values is not zero and is attained at the point c then c ( a, b ) and, by the previous theorem f 0 ( c ) = 0. Mean Value Theorem. If f : [ a, b ] R is continuous and it is differentiable at every x ( a, b ) then there exists a point c in ( a, b ) at which f 0 ( c ) = f ( b ) - f ( a ) b - a . Proof. Consider the function g ( x ) = f ( x ) - f ( a ) - f ( b ) - f ( a ) b - a ( x - a ) . Since f is continuous on [ a, b ] and differentiable on ( a, b ), the same is true about g . Also, g ( a ) = g ( b ) = 0. Applying Rolle’s Theorem to g , we obtain the required result. Corollary. If f is differentiable on an interval ( a, b ) and f 0 ( x ) = 0 for all x ( a, b ) then f is constant on ( a, b ). Proof. Let a 1 , b 1 ( a, b ) and a 1 < b 1 . Applying Mean Value Theorem to the interval [ a 1 , b 1 ], we obtain f ( b 1 ) - f ( a 1 ) b 1 - a 1 = 0, that is, f ( b 1 ) = f ( a 1 ). Since this is true for all a 1 , b 1 ( a, b ), the function f is constant.
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