Applications of Derivatives
Monotonic Functions
Grade 12
Question:
<p>Let \(f(x) = \int_x^{x^3} \frac{dt}{2\ln t}\) for \(x > 1\) and \(g(x) = \int_1^x (2t^2 - \ln t)f(t)dt\) \((x > 1)\), then:</p>
<p>(a) g is increasing on \((1, \infty)\)</p>
<p>(b) g is decreasing on \((1, \infty)\)</p>
<p>(c) g is increasing on \((1, 2)\) and decreasing on \((2, \infty)\)</p>
<p>(d) g is decreasing on \((1, 2)\) and increasing on \((2, \infty)\)</p>
Step-by-Step Solution
Key Concept: Use Leibniz integral rule to differentiate g(x) and analyze the sign of the derivative.
<p>Find $g'(x) = (2x^2 - \ln x)f(x)$ using Leibniz rule. Analyze the sign of $g'(x)$ to determine monotonicity.</p>
Correct Answer: c