Lagrange ’ s Mean Value Theorem
Let f be a real function , continuous on the closed interval [ a , b ] and differentiable in the open interval ( a , b ). Then , there is at least one point c in the open interval ( a , b ), such that
Geometrically Any chord of the curve y = f ( x ), there is a point on the graph , where the tangent is parallel to this chord .
Remarks In the particular case , where f ( a ) = f ( b ). The expression [ f ( b ) – f ( a )/( b – a )] becomes zero . Thus , when f ( a ) = f ( b ), f ‗ ( c ) = 0 for some c in ( a , b ).
Thus , Rolle ‘ s theorem becomes a particular case of the mean value theorem .
Approximations and Errors
1 . Let y = f ( x ) be a given function . Let Ax denotes a small increment in Δx , corresponding which y increases by Δy . Then , for small increments , we assume that