Q. 94.0( 57 Votes )

# Prove that the function f given by f (x) = | x – 1|, x ∈ R is not differentiable at x = 1.

Given: f(x)=|x-1|, x R

because a function f is differentiable at a point x=c in its domain if both its limits as:

are finite and equal.

Now, to check the differentiability of the given function at x=1,

Let we consider the left hand limit of function f at x=1

because, {h < 0 |h|= -h}

= -1

Now, let we consider the right hand limit of function f at x=1

because, {h>0 |h|= h}

= 1

Because, left hand limit is not equal to right hand limit of function f at x=1, so f is not differentiable at x=1.

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