At which point does f(x) = |x – 1| has itslocal minimum?

Category: QuestionsAt which point does f(x) = |x – 1| has itslocal minimum?
Editor">Editor Staff asked 11 months ago

At which point does f(x) = |x – 1| has itslocal minimum?
 
(a) They are unequal
 
(b) They are equal
 
(c) Depend on the numbers
 
(d) Can’t be predicted
 
The question was asked in examination.
 
My doubt stems from Calculus Application in section Application of Calculus of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience

1 Answers
Editor">Editor Staff answered 11 months ago

Right answer is (b) They are equal
 
To explain I would say: 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

hf(c + h) – f(c)
 
And
 
lim
h→
0
+
hf(c + h) – f(c) are finite and equal.
 
To check the differentiability of the function at x = 1,
 
LHS,
 
Consider the left hand limit of f at x=1
 
lim
h→
0

|1+h−1|−|1−1|
h
 
 
=
lim
h→
0

|h|
h
 
 
=
lim
h→
0

−h
h
 
 
= −1
 
RHS,
 
Consider the right hand limit of f at x − 1
 
lim
h→
0
+
|1+h−1|−|1−1|
h
 
 
=
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.
 
As, LHS = -1 and RHS = 1, it is clear that, f’(1) < 0 on the left of x = 1 and f’(x) > 0 on the right of the point x = 1.
 
Hence, f’(x) changes sign, from negative on the left to positive on the right of the point x = 1.
 
Therefore, f(x) has a local minima at x = 1.