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Question
Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?
Solution
Given:
\[f\left( x \right) = \left| x \right|\]
If Lagrange's theorem is applicable for the given function, then \[f\left( x \right)\] is continuous on \[\left[ - 1, 1 \right]\] and differentiable on \[\left( - 1, 1 \right)\] But it is known that \[f\left( x \right) = \left| x \right|\] is not differentiable at \[x = 0 \in \left( - 1, 1 \right)\] .
Thus, our supposition is wrong.
Therefore, Lagrange's theorem is not applicable for the given function.
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