If \( p(x) = f(x)g(x) \), then which of the following statements is/are correct ? I. \( p(x) \) is continuous at x = 0. II. \( p(x) \) is differentiable at x = 0. Select the answer using the code given below :
For the next two (02) items that follow : Let \( f(x) = \tan(x^2) \) and \( g(x) = x|x| \) for \( |x| < \sqrt{\frac{\pi}{2}} \).
- A. I only
- B. II only
- C. Both I and II ✓
- D. Neither I nor II
Correct Answer: C. Both I and II
Explanation
The function \( p(x) = x|x|\tan(x^2) \). Near x=0, \( |p(x)| \approx |x^4| \), which approaches 0, ensuring continuity. The derivative at x=0 evaluated from both sides is also 0. Thus, it is both continuous and differentiable.
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