Consider the following statements in respect of f(x) = |x| - 1

1. f(x) is continuous at x = 1.

2. f(x) is differentiable at x = 0.

Which of the above statements is/are correct?

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NDA 02/2021: Maths Previous Year paper (Held On 14 Nov 2021)
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  1. 1 only
  2. 2 only
  3. Both 1 and 2
  4. Neither 1 nor 2

Answer (Detailed Solution Below)

Option 1 : 1 only
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Detailed Solution

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Concept:

Differentiable Functions:

  • If a graph has a sharp corner at a point, then the function is not differentiable at that point.
  • If a graph has a break at a point, then the function is not differentiable at that point.
  • If a graph has a vertical tangent line at a point, then the function is not differentiable at that point.

 

Calculation:

Givne that,

f(x) = |x| - 1     ----(1)

Step: 1

Step: 2 

Step: 3 

Clearly, we can see that,

f(x) is continuous at x = 1 and

f(x) is not differentiable at x = 0 because there is a corner at x = 0.

∴ Only statement 1 is correct.

Additional Information 

  • Differentiable functions are those functions whose derivatives exist.
  • If a function is differentiable, then it is continuous.
  • If a function is continuous, then it is not necessarily differentiable.
  • The graph of a differentiable function does not have breaks, corners, or cusps.

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