Applications of Derivatives
Monotonicity of Trigonometric Functions
Grade 12
Question:
<p>The function \(f(x) = \sin^4 x + \cos^4 x\) is increasing, if</p>
<p>(a) \(0 < x < \frac{7\pi}{8}\)</p>
<p>(b) \(\frac{7\pi}{4} < x < \frac{3\pi}{8}\)</p>
<p>(c) \(\frac{3\pi}{8} < x < \frac{5\pi}{8}\)</p>
<p>(d) \(\frac{5\pi}{8} < x < \frac{3\pi}{4}\)</p>
Step-by-Step Solution
Key Concept: Differentiate using the chain rule, simplify using trigonometric identities, and solve $f'(x) > 0$ to find increasing intervals.
<p><strong>Step 1:</strong> Find the derivative: $f'(x) = 4\sin^3 x \cos x + 4\cos^3 x(-\sin x)$</p><p><strong>Step 2:</strong> Factor: $f'(x) = 4\sin x \cos x(\sin^2 x - \cos^2 x) = 2\sin 2x(-\cos 2x) = -\sin 4x$</p><p><strong>Step 3:</strong> For $f'(x) > 0$: $-\sin 4x > 0 \Rightarrow \sin 4x < 0$</p><p>This occurs when $\pi < 4x < 2\pi$, i.e., $\frac{\pi}{4} < x < \frac{\pi}{2}$</p><p>The interval $\left[\frac{3\pi}{8}, \frac{5\pi}{8}\right)$ is a subset of $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$.</p><p><strong>∴ Answer is (b)</strong></p>
Correct Answer: b