MHT CET · Maths · Application of Derivatives
The function \(\mathrm{f}(x)=\sin ^4 x+\cos ^4 x\) increases if
- A \(0 < x < \frac{\pi}{8}\)
- B \(\frac{\pi}{4} < x < \frac{\pi}{2}\)
- C \(\frac{3 \pi}{8} < x < \frac{5 \pi}{8}\)
- D \(\frac{5 \pi}{8} < x < \frac{3 \pi}{4}\)
Answer & Solution
Correct Answer
(B) \(\frac{\pi}{4} < x < \frac{\pi}{2}\)
Step-by-step Solution
Detailed explanation
\(\mathrm{f}(x)=\sin ^4 x+\cos ^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - \frac{1}{2}(2\sin x \cos x)^2 = 1 - \frac{1}{2}\sin^2(2x)\) \(\mathrm{f}(x) = 1 - \frac{1}{2} \left( \frac{1-\cos(4x)}{2} \right) = 1 - \frac{1}{4} + \frac{1}{4}\cos(4x) = \frac{3}{4} + \frac{1}{4}\cos(4x)\)
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