1.

Consider the following statements:1. f(x) = |x - 3| is continuous at x = 02. f(x) = |x - 3| is differentiable at x = 0Which of the statements given above is/are correct?1. 1 only2. 2 only3. Both 1 and 24. Neither 1 nor 2

Answer» Correct Answer - Option 3 : Both 1 and 2

Concept:

|x| = x if x ≥  0 and |x| = -x , if x < 0

We say f(x) is continuous at x = a if \(\rm\lim _{x \rightarrow a^{-}} f(x)=\lim _{x \rightarrow a^{+}} f(x)=lim _{x \rightarrow a} f(x)\)

We say f(x) is differentiable at x = a if \(\rm\lim _{x \rightarrow a^{-}} f'(x)=\lim _{x \rightarrow a^{+}} f'(x)\)

 

Calculation:

Here,  f(x) = |x - 3|

f(x) = x - 3 when x - 3 ≥  0 ⇒ x ≥ 3

And f(x) = 3 - x when x - 3 < 0 ⇒ x < 3

For, x = 0, f(x) = 3 - x

Function is not changing at x = 0   (∵ function is linear)

So, f(x) is continuous at x = 0

Now, x = 0, f(x) = 3 - x

⇒ f'(x) = -1

f'(0-) = -1 and f'(0+) = -1

So, f(x) is differentiable at x = 0

Hence, option (3) is correct.


Discussion

No Comment Found

Related InterviewSolutions