Tamil Nadu Board of Secondary EducationHSC Arts Class 11

# Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1? f(x)=|x-1| - Mathematics

Sum

Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

f(x) = |x - 1|

#### Solution

f(x) = {{:(x - 1,  "if"  x > 1),(-(x - 1),  "if"  x < 1):}

First we find left limit of f(x) at x = 1

When x -> 1- we have f(x) = - (x - 1)

f"'"(1^-) =  lim_(x -> 1^-) (f(x) - f(1))/(x - 1)

f"'"(1^-) =  lim_(x -> 1) (-(x - 1) - (0))/(x - 1)

= lim_(x -> 1) (-(x - 1))/(x - 1) = – 1  .......(1)

f"'"(1^+) =  lim_(x -> 1^+) (f(x) - f(1))/(x - 1)

= lim_(x -> 1^+) ((x - 1) - 0)/(x - 1)

f"'"(1^+) =  lim_(x -> 1^+) (x - 1)/(x - 1) = 1  ......(2)

From equation (1) and (2) we have

lim_(x -> 1^-) f(x)  ≠  lim_(x -> 1^+) f(x)

∴ f"'"(x) does not exist at x = 1

Concept: Differentiability and Continuity
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Chapter 10: Differential Calculus - Differentiability and Methods of Differentiation - Exercise 10.1 [Page 147]

#### APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 11th Mathematics Volume 1 and 2 Answers Guide
Chapter 10 Differential Calculus - Differentiability and Methods of Differentiation
Exercise 10.1 | Q 2. (i) | Page 147
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