Definite Integration
Leibniz Rule for Integrals
Grade 12
Question:
<p>On the interval <\(\frac{\pi}{8}, \frac{p}{4}\)>, find the least value of the function <\(f(x) = \int_{5x}^{\frac{\pi}{4}} (3\sin t - 4\cos t) dt\)>.</p>
<p>(a) <\(-\frac{3}{2} + 2\sqrt{3}\)></p>
<p>(b) <\(\frac{3}{2} - 2\sqrt{3}\)></p>
<p>(c) <\(\frac{3}{2} + 2\sqrt{3}\)></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use Leibniz rule for differentiation under the integral sign to find critical points, then evaluate the function at critical points and endpoints.
<p><strong>Solution:</strong> Given <$f(x) = \int_{5x}^{\frac{\pi}{4}} (3\sin t - 4\cos t) dt$></p><p>By Leibniz rule: <$f'(x) = 3\sin x - 4\cos x$> for <$x \in \left[\frac{\pi}{8}, \frac{\pi}{4}\right]$></p><p>Hence, (d) is the correct answer.</p>
Correct Answer: d