What is the <strong>MAXIMUM</strong> value of 3(\sin x-\cos x)+4(\cos^{3}x-\sin^{3}x) ?
- A. 1
- B. \sqrt{2} ✓
- C. \sqrt{3}
- D. 2
Correct Answer: B. \sqrt{2}
Explanation
Rearrange the expression grouping sine and cosine terms: (3\sin x - 4\sin^3 x) - (3\cos x - 4\cos^3 x). Using the multiple-angle identities \sin 3x = 3\sin x - 4\sin^3 x and \cos 3x = 4\cos^3 x - 3\cos x, the expression simplifies to \sin 3x + \cos 3x. The maximum value of a\sin \theta + b\cos \theta is \sqrt{a^2+b^2}. Here, a=1, b=1, so the maximum value is \sqrt{1^2+1^2} = \sqrt{2}.