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NDA 1 2024 Maths, question 87

Topic: Differential calculus

Passage · Q87 to Q88

Let f(x)=∣x∣+1f(x)=|x|+1 and g(x)=[x]−1g(x)=[x]-1, where [⋅][\cdot ] is the greatest integer function.

Let h(x)=f(x)g(x)h(x)=\frac{f(x)}{g(x)}

Q87

Consider the following statements :

  1. f(x)f(x) is differentiable for all x<0x<0

  2. g(x)g(x) is continuous at x=0.0001x=0.0001

  3. The derivative of g(x)g(x) at x=2.5x=2.5 is 11

Which of the statements given above are correct ?

  1. (a) 1 and 2 onlyCorrect answer
  2. (b) 2 and 3 only
  3. (c) 1 and 3 only
  4. (d) 1, 2 and 3

Solution.

Answer: (a) 1 and 2 only

  1. For x < 0, |x| + 1 = 1 − x, which is differentiable.
  2. The floor function is constant near the noninteger 0.0001 and is continuous there.
  3. It is also constant near 2.5, so its derivative is 0, not 1.
  4. Only statements 1 and 2 hold.