Applications of Derivatives
Critical Points and Extrema
Grade 12
Question:
<p>The number of critical points of \(f(x) = \left(\int_0^x (\cos^3 t - \frac{3}{4}t^{4/3})dt\right)^2 + 4x - 2\) in \([0, 6\pi]\) is:</p>
<p>(a) 10</p>
<p>(b) 8</p>
<p>(c) 6</p>
<p>(d) 12</p>
Step-by-Step Solution
Key Concept: Use chain rule and Leibniz rule to find f'(x), then count roots in the given interval.
<p>Let $u(x) = \int_0^x (\cos^3 t - \frac{3}{4}t^{4/3})dt$. Then $f(x) = u(x)^2 + 4x - 2$. We have $f'(x) = 2u(x)u'(x) + 4 = 2u(x)(\cos^3 x - \frac{3}{4}x^{4/3}) + 4$. Critical points occur when $f'(x) = 0$.</p>
Correct Answer: b