JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(f(x)=3 \sin ^{4} x+10 \sin ^{3} x+6 \sin ^{2} x-3, x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right] .\) Then, \(f\) is \(.....\)
- A increasing in \(\left(-\frac{\pi}{6}, 0\right)\)
- B decreasing in \(\left(0, \frac{\pi}{2}\right)\)
- C decreasing in \(\left(-\frac{\pi}{6}, 0\right)\)
- D increasing in \(\left(-\frac{\pi}{6}, \frac{\pi}{2}\right)\)
Answer & Solution
Correct Answer
(C) decreasing in \(\left(-\frac{\pi}{6}, 0\right)\)
Step-by-step Solution
Detailed explanation
\(f(x)=3 \sin ^{4} x+10 \sin ^{3} x+6 \sin ^{2} x-3, x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]\) \(f(x)=12 \sin ^{3} x \cos x+30 \sin ^{2} x \cos x+12 \sin x \cos x\) \(=6 \sin x \cos x\left(2 \sin ^{2} x+5 \sin x+2\right)\) \(=6 \sin x \cos x(2 \sin x+1)(\sin +2)\)…
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