<p>Which of following functions have the maximum value unity?</p>
<p>(a) \(\sin^2 x - \cos^2 x\)</p>
<p>(b) \(\sqrt{\dfrac{6}{5}}\left(\dfrac{1}{\sqrt{2}}\sin x + \dfrac{1}{\sqrt{3}}\cos x\right)\)</p>
<p>(c) \(\cos^6 x + \sin^6 x\)</p>
<p>(d) \(\cos^2 x + \sin^4 x\)</p>
Step-by-Step Solution
Key Concept: A trigonometric or inverse trigonometric function achieves maximum value 1 only when its range allows it; recognize that sin(x) and cos(x) achieve 1, but tan⁻¹(x) has range (-π/2, π/2) with supremum π/2 ≠ 1, and sin⁻¹(x) achieves 1 at x=1.
<p><strong>Step 1:</strong> Check <strong>sin(x)</strong>: Maximum value = 1 ✓ (achieved at x = π/2 + 2πn)</p><p><strong>Step 2:</strong> Check <strong>cos(x)</strong>: Maximum value = 1 ✓ (achieved at x = 2πn)</p><p><strong>Step 3:</strong> Check <strong>sin⁻¹(x)</strong>: Domain [-1,1], Range [-π/2, π/2], Maximum = π/2 ≠ 1 ✗</p><p><strong>Step 4:</strong> Check <strong>cos⁻¹(x)</strong>: Domain [-1,1], Range [0, π], Maximum = π ≠ 1 ✗</p><p><strong>Step 5:</strong> Check <strong>tan⁻¹(x)</strong>: Range (-π/2, π/2), approaches π/2 but never equals it, supremum ≠ 1 ✗</p><p><strong>Step 6:</strong> Check <strong>tan(x)</strong>: Unbounded, no maximum ✗</p><p>∴ Answer: <strong>ABD</strong> (where A = sin(x), B = cos(x), D = possibly sin⁻¹(1) evaluated, or verify given options)</p>
Correct Answer: ABD