<p>The function \(f(x) = \sin^3 x - m \sin x\) is defined on open interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) and if assumes only 1 maximum value and only 1 minimum value on this interval. Then, which one of the following must be correct?</p>
Step-by-Step Solution
Key Concept: For f(x) to have exactly one maximum and one minimum on the open interval, its derivative f'(x) must have exactly two zeros in this interval where f changes from increasing to decreasing (or vice versa). We need to analyze when f'(x) = 3sin²x - m changes sign exactly twice.
<p><strong>Step 1: Find the derivative.</strong></p><p>f(x) = sin³x - m sin x</p><p>f'(x) = 3sin²x cos x - m cos x = cos x(3sin²x - m)</p><p></p><p><strong>Step 2: Identify critical points.</strong></p><p>f'(x) = 0 when cos x = 0 or 3sin²x - m = 0</p><p>In the open interval (-π/2, π/2):</p><p>• cos x = 0 only at x = 0 (boundary points excluded)</p><p>• 3sin²x = m requires m ≥ 0 and sin²x = m/3</p><p></p><p><strong>Step 3: Analyze the condition for exactly one max and one min.</strong></p><p>For exactly one maximum and one minimum in (-π/2, π/2), we need f'(x) = 0 to have exactly two interior zeros that correspond to extrema.</p><p>Since cos x ≠ 0 in the open interval (except would be at boundaries), the zeros come from 3sin²x = m.</p><p></p><p><strong>Step 4: Determine conditions on m.</strong></p><p>For sin²x = m/3 to have two solutions in (-π/2, π/2):</p><p>• We need 0 < m/3 < 1, so 0 < m < 3</p><p>• If m/3 = a where 0 < a < 1, then sin x = ±√a gives exactly two values in the open interval</p><p>• One solution is positive (gives a max) and one is negative (gives a min)</p><p></p><p><strong>Step 5: Verify boundary behavior.</strong></p><p>As x → ±π/2, sin x → ±1, so f(x) → 1 - m or -1 + m.</p><p>For interior extrema to be the only extrema in this open interval, we need 0 < m < 3.</p><p>If m ≤ 0: f'(x) = cos x(3sin²x - m) ≥ 0, so f is monotonic (no interior extrema).</p><p>If m ≥ 3: sin²x = m/3 ≥ 1 is impossible for real x.</p><p></p><p><strong>∴ Answer:</strong> a</p>
Correct Answer: a