Q. 975.0( 1 Vote )

# Fill in the blanks in each of theAn example of a function which is continuous everywhere but fails to be differentiable exactly at two points is _______.

Consider, f(x) = |x-1| + |x-2|

Let’s discuss the continuity of f(x).

We have, f(x) = |x-1| + |x-2|

When x<1, we have f(x) = -2x+3, which is a polynomial function and polynomial function is continuous everywhere.

When 1≤x<2, we have f(x) = 1, which is a constant function and constant function is continuous everywhere.

When x≥2, we have f(x) = 2x-3, which is a polynomial function and polynomial function is continuous everywhere.

Hence, f(x) = |x-1| + |x-2| is continuous everywhere.

Let’s discuss the differentiability of f(x) at x=1 and x=2.

We have

Lf’(1) =

=

= ( f(x) = -2x+3, if x< 1)

=

=

Rf’(1) =

= ( f(x) = 1, if 1≤ x< 2)

=0

Lf’(1) ≠ Rf’(1)

f(x) is not differentiable at x=1.

Lf’(2) =

= (f(x) = 1, if 1≤ x< 2 and f(2) = 2×2-3 =1 )

=0

Rf’(2) =

=

= ( f(x) = 2x-3, if x≥ 2)

=

=

Lf’(2) ≠ Rf’(2)

f(x) is not differentiable at x=2.

Thus, f(x) = |x-1| + |X-2| is continuous everywhere but fails to be differentiable exactly at two points x=1 and x=2.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Super 10 Question: Check Your Knowledge of Maxima & Minima (Quiz)45 mins
Maxima & Minima in an interval60 mins
Questions Based on Maxima & Minima57 mins
Questions based on Maxima & Minima in an interval59 mins
Check your Knowlege of Maxima & Minima ( Challenging Quiz)60 mins
Connection B/w Continuity & Differentiability59 mins
Problems Based on L-Hospital Rule (Quiz)0 mins
When does a Maxima or Minima occur?48 mins
Interactive Quiz | Differentiability by using first principle59 mins
Interactive Quiz on Limits67 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses