Under which one of the following conditions does the function f(x)=(p~\sec~x)^{2}+(q~\text{cosec}~x)^{2} attain <strong>MINIMUM</strong> value ?
- A. \tan^{2}x=\frac{q}{p} ✓
- B. \cot^{2}x=\frac{q}{p}
- C. \tan^{2}x=pq
- D. \cot^{2}x=pq
Correct Answer: A. \tan^{2}x=\frac{q}{p}
Explanation
Rewrite the function as f(x) = p^2(1+\tan^2 x) + q^2(1+\cot^2 x) = p^2 + q^2 + p^2\tan^2 x + q^2\cot^2 x. By AM-GM inequality, p^2\tan^2 x + q^2\cot^2 x \geq 2pq. The minimum is attained when p^2\tan^2 x = q^2\cot^2 x, which implies \tan^4 x = \frac{q^2}{p^2} \implies \tan^2 x = \frac{q}{p}.
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