Points of Non-Differentiability

Calculus Level 5

f ( x ) = x 2 − ∣ x 2 − 1 ∣ + 2 ∣ ∣ x ∣ − 1 ∣ + 2 ∣ x ∣ − 7 f(x)=x^2-\mid x^2-1\mid+2\mid\mid x\mid-1\mid+2\mid x\mid-7

Number of points where the above function is non-differentiable is?

Try for some more interesting problems of Limits and Derivatives.


The answer is 0.

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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2 solutions

Gagan Raj
Feb 15, 2015

The given function is differentiable at all points (Verified Graphically)

Hence f(x) has zero points of non - differentiability.

Hence the answer is 0.

Enjoy and Learn!!!!!

6 Helpful 0 Interesting 0 Brilliant 0 Confused

How do you know it is differentiable at all points?

Pi Han Goh - 6 years, 3 months ago

Yep I too have verified graphically. Can someone tell me how to do above question without using graph.

A Former Brilliant Member - 6 years, 3 months ago

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the proper way to solve this problem at those particular points, is as follows:
Say we take x = 1 x = 1 .
The Right hand derivative is 2 x − 2 x + 2 x + 2 x = 4 x 2x - 2x + 2x + 2x = 4x , which evaluates to 4.
The Left hand derivative is 2 x − ( − 2 x ) − 2 x + 2 x = 4 x 2x - (-2x) -2x + 2x = 4x , which evaluates to 4.
Hence the derivative at x = 1 x = 1 is 4.
Now, repeat this with x = 0 , − 1 x = 0, -1 .


Calvin Lin Staff - 6 years, 2 months ago

U can analyse for different cases and find it's gradient and you would see it cancels out

Julian Poon - 6 years, 3 months ago
Saharsh Rathi
Nov 4, 2016

We only need to check differentiability at the end points .

5 Helpful 0 Interesting 0 Brilliant 0 Confused

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