Q2 of 96 Page 15

Discuss the applicability of Lagrange’s mean value theorem for the function f(x) = |x| on [ – 1, 1].

Lagrange’s mean value theorem 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 is also known as First Mean Value Theorem.


f(x) = |x| on [ – 1, 1]



For differentiability at x=0,



{Since f(x)= – x, x<0}





= – 1



{Since f(x)= x, x>0}





= 1



⇒ f(x) is not differential at x=0


∴ Lagrange’s mean value theorem is not applicable for the function f(x) = |x| on [ – 1, 1].


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