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NDA 1 2025 Maths, question 80

Topic: Differential calculus

Passage · Q79 to Q80

Let f(x)={x3,x2<1,x2,x2≥1.f(x) = \left\{ \begin{matrix} x^{3}, & x^{2} < 1, \\ x^{2}, & x^{2} \geq 1. \end{matrix} \right.

Q80

Consider the following statements: I. The function is continuous at x=−1x = - 1. II. The function is differentiable at x=1x = 1. Which of the statements given above is/are correct?

  1. (a) I only
  2. (b) II only
  3. (c) Both I and II
  4. (d) Neither I nor IICorrect answer

Solution.

Answer: (d) Neither I nor II

  1. At x=−1x = - 1, the left-hand limit comes from x2x^{2} and equals 1, while the right-hand limit comes from x3x^{3} and equals −1- 1.
  2. Hence the function is not continuous there, so I is false.
  3. At x=1x = 1, both branches give value 1, but the left derivative is 33 and the right derivative is 22.
  4. Thus the function is not differentiable there, and II is false.