CUET · MATHS · PYQ PAPER 2023
The interval in which the function given by \(f(x) = \sin^4 x + \cos^4 x\), x \(\in\) [0, \(\frac{\pi}{2}\)] is decreasing is:
- A \(\left(0, \frac{\pi}{4}\right]\)
- B \(\left(0, \frac{\pi}{4}\right)\)
- C \(\left[0, \frac{\pi}{4}\right)\)
- D \(\left[0, \frac{\pi}{4}\right]\)
Answer & Solution
Correct Answer
(B) \(\left(0, \frac{\pi}{4}\right)\)
Step-by-step Solution
Detailed explanation
\(f(x) = \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - \frac{1}{2}(2\sin x \cos x)^2 = 1 - \frac{1}{2}\sin^2 2x\) \(f'(x) = 0 - \frac{1}{2} \cdot 2\sin 2x \cdot \cos 2x \cdot 2 = -2\sin 2x \cos 2x = -\sin 4x\) For \(f(x)\) to be decreasing, \(f'(x) 0\)…
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