Consider the following statements : I. The function is continuous at x=-1. II. The function is differentiable at x=1. Which of the statements given above is/are correct?
Consider the following for the two (02) items that follow : Let f(x)=\begin{cases}x^{3},&x^{2}\lt 1\\ x^{2},&x^{2}\geq 1\end{cases}
- A. I only
- B. II only
- C. Both I and II
- D. Neither I nor II ✓
Correct Answer: D. Neither I nor II
Explanation
For statement I: At x=-1, the left-hand limit (x \to -1^-, where x^2 \gt 1) is (-1)^2 = 1. The right-hand limit (x \to -1^+, where x^2 \lt 1) is (-1)^3 = -1. Since 1 \neq -1, f(x) is not continuous at x=-1. For statement II: At x=1, LHD (x \to 1^-) uses f'(x) = 3x^2 \implies 3(1)^2 = 3. RHD (x \to 1^+) uses f'(x) = 2x \implies 2(1) = 2. Since LHD \neq RHD, f(x) is not differentiable at x=1. Both are incorrect.
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