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Consider the function f(x)=[(1-sin³ x)/(3cos² x),, x<(π)/(2),; [4pt] p,, x=(π)/(2),; [4pt] q(1-sin x),, x>(π)/(2).] If f is continuous at x=(π)/(2), then the pair (p,q) is

1. (1, 4)
2. (1/2, 2)
3. (1/2, 4)
4. None of these

Correct answer: (1/2, 4)

Solution

As x->pi/2 from the left, (1-sin^3 x)/(3cos^2 x) -> 1/2 (factor 1-sin^3 x = (1-sin x)(1+sin x+sin^2 x) and cos^2 x = 1-sin^2 x), so p = 1/2. Matching the right-hand piece q(1-sin x)/(pi-2x)^2 -> q/8 = 1/2 gives q = 4. Hence (p,q) = (1/2, 4).

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