NDA 2023 Maths · Mathematics · Continuity and Differentiability · hard

The function f(x)=|x-1| is differentiable at x=1. 2. If f is continuous on [a,b], then f'(c)=0 for some c∈(a,b).

Single correct option

Consider the following statements:
1. The function

f(x)=|x-1|

is differentiable at

x=1

.
2. If

f

is continuous on

[a,b]

, then

f'(c)=0

for some

c\in(a,b)

.
Which of the statements given above is/are correct?

Show answer and solution

Answer: (D) Neither 1 nor 2

x=1

the left derivative is

-1

and the right derivative is

+1

, so

|x-1|

is not differentiable there; Statement 1 is false. Statement 2 is not true: Rolle's theorem also needs differentiability on

(a,b)

and

f(a)=f(b)

. For example

f(x)=x

[0,1]

is continuous but

f'

never vanishes.

Original question written in the NDA 2023 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake