If \( q(x) = f \circ g(x) \), then which of the following statements is/are correct ? I. \( q(x) \) is continuous at x = 0. II. \( q(x) \) 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 composition \( q(x) = f(g(x)) = \tan((x|x|)^2) = \tan(x^4) \). This function behaves smoothly near x=0. Its limit at 0 is 0 (continuous) and its derivative \( 4x^3 \sec^2(x^4) \) is 0 at x=0 (differentiable).
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