Question

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

Answer 1

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Answer 2

Step-by-step explanation:

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Continuity and Differentiability

Differentiability of a Function

Prove that the function f g...

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Asked on November 22, 2019 by

Neelam Samariya

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

EASY

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ANSWER

The given function is f(x)=∣x−1∣,x∈R.

It is known that a function f is differentiable at point x=c in its domain if both

lim

h→0

−

h

f(c+h)−f(c)

and lim

h→0

+

h

f(c+h)−f(c)

are finite and equal.

To check the differentiability of the function at x=1,

Consider the left hand limit of f at x=1

lim

h→0

−

h

∣1+h−1∣−∣1−1∣

=lim

h→0

−

h

∣h∣

=lim

h→0

−

h

−h

=−1

Consider the right hand limit of f at x−1

lim

h→0

+

h

∣1+h−1∣−∣1−1∣

=lim

h→0

+

h

h

=1

Since the left and right hand limits of f at x=1 are not equal, f is not differentiable at x=1.