Applications of Derivatives
Increasing and Decreasing Functions
Grade 12
Question:
<p>The interval in which \(f(x) = 3\cos^4 x + 10\cos^3 x + 6\cos x - 3\) decreases is \(x \in [0, \pi]\)</p>
<p>(a) \(\left(0, \frac{\pi}{2}\right) \cup \left(\frac{2\pi}{3}, \pi\right)\)</p>
<p>(b) \(\left(\frac{\pi}{2}, \frac{2\pi}{3}\right)\)</p>
<p>(c) \(\left(0, \frac{\pi}{3}\right) \cup \left(\frac{2\pi}{3}, \pi\right)\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Find where the derivative is negative to identify the decreasing intervals.
<p>Let $u = \cos x$. Then $f(x) = 3u^4 + 10u^3 + 6u - 3$.</p><p>$\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = (12u^3 + 30u^2 + 6)(-\sin x)$</p><p>Setting $12u^3 + 30u^2 + 6 = 0$ and solving for critical points in $[0, \pi]$.</p><p>$f$ decreases when $\frac{df}{dx} < 0$, which occurs on $\left(0, \frac{\pi}{2}\right) \cup \left(\frac{2\pi}{3}, \pi\right)$</p>
Correct Answer: A