Trigonometry & Inverse Trigonometry
Maxima and Minima
Grade 11
Question:
<p>If \(a > \frac{1}{\sin^6 x + \cos^6 x}\) for all \(x \in \mathbb{R}\), then \(a\) can be:</p>
<p>(a) \(3\)</p>
<p>(b) \(4\)</p>
<p>(c) \(5\)</p>
<p>(d) \(6\)</p>
Step-by-Step Solution
Key Concept: Find the minimum value of \(\sin^6 x + \cos^6 x\) to determine the maximum value of its reciprocal.
<p>We need to find the maximum value of \(\frac{1}{\sin^6 x + \cos^6 x}\). Using the identity \(\sin^6 x + \cos^6 x = 1 - 3\sin^2 x \cos^2 x = 1 - \frac{3}{4}\sin^2 2x\). The minimum of \(\sin^6 x + \cos^6 x\) is \(\frac{1}{4}\) (when \(\sin 2x = \pm 1\)). Thus the maximum of \(\frac{1}{\sin^6 x + \cos^6 x}\) is \(4\). So \(a > 4\), meaning \(a \in \{5, 6\}\).</p>
Correct Answer: c, d