<p>The function \(f(x) = \int_0^x t(e^t - 1)(t - 1)(t - 2)^3(t - 3)^5 dt\) has a local maximum at <i>x</i> equals</p>
Step-by-Step Solution
Key Concept: Use Leibniz integral rule to find the derivative: f'(x) = integrand evaluated at x. Analyze sign changes considering the multiplicities of factors to identify local maxima.
<p><strong>Solution:</strong></p><p>Given: $f(x) = \int_0^x t(e^t - 1)(t - 1)(t - 2)^3(t - 3)^5 dt$</p><p>By Leibniz rule: <i>f</i>'(<i>x</i>) = <i>x</i>(<i>e</i><sup><i>x</i></sup> - 1)(<i>x</i> - 1)(<i>x</i> - 2)<sup>3</sup>(<i>x</i> - 3)<sup>5</sup></p><p>Analyzing the sign of <i>f</i>'(<i>x</i>) around critical points using the number line rule and considering the multiplicities of roots, a local maximum occurs at <i>x</i> = 3.</p><p>∴ Answer is (d) 3</p>
Correct Answer: d