Question

give 1 example of a function and prove that function f is differentiable at x= x0.

You may use "insert equations"

Answer 1

Answer:

umm sorry I m not good at maths

Answer 2

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.