Question

prove that function f given by f (x)=|x_1|,x€r is not differentiable at x = 1​

Answer 1

Step-by-step explanation:

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

limh→0−hf(c+h)−f(c) and limh→0+hf(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

limh→0−h∣1+h−1∣−∣1−1∣=limh→0−h∣h∣=limh→0−h−h=−1

Consider the right hand limit of f at x−1

limh→0+h∣1+h−1∣−∣1−1∣=limh→0+hh=1

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

Answer 2

The given function is f(x)=∣x−1∣,x∈R.

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

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 bothlimh→0−hf(c+h)−f(c) and limh→0+hf(c+h)−f(c) are finite and equal.

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 bothlimh→0−hf(c+h)−f(c) and limh→0+hf(c+h)−f(c) are finite and equal.To check the differentiability of the function at x=1,

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 bothlimh→0−hf(c+h)−f(c) and limh→0+hf(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

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 bothlimh→0−hf(c+h)−f(c) and limh→0+hf(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=1limh→0−h∣1+h−1∣−∣1−1∣=limh→0−h∣h∣=limh→0−h−h=−1

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 bothlimh→0−hf(c+h)−f(c) and limh→0+hf(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=1limh→0−h∣1+h−1∣−∣1−1∣=limh→0−h∣h∣=limh→0−h−h=−1Consider the right hand limit of f at x−1

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 bothlimh→0−hf(c+h)−f(c) and limh→0+hf(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=1limh→0−h∣1+h−1∣−∣1−1∣=limh→0−h∣h∣=limh→0−h−h=−1Consider the right hand limit of f at x−1limh→0+h∣1+h−1∣−∣1−1∣=limh→0+hh=1

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 bothlimh→0−hf(c+h)−f(c) and limh→0+hf(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=1limh→0−h∣1+h−1∣−∣1−1∣=limh→0−h∣h∣=limh→0−h−h=−1Consider the right hand limit of f at x−1limh→0+h∣1+h−1∣−∣1−1∣=limh→0+hh=1Since the left and right hand limits of f at x=1 are not equal, f is not differentiable at x=1.

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