Definite Integration
Application of Leibniz Rule
Grade 12
Question:
<p>If \(m\) and \(n\) are positive integers and \(\int_a^b (t - a)^{2n} ((t - b)^{2m+1} \, dt\), \(a < b\), then</p>
<p>(A) \(x = b\) is a point of local minimum</p>
<p>(B) \(x = b\) is a point of local maximum</p>
<p>(C) \(x = a\) is a point of local minimum</p>
<p>(D) \(x = a\) is a point of local maximum</p>
Step-by-Step Solution
Key Concept: Apply Leibniz rule for differentiation under the integral sign to analyze critical points.
<p>The integrand is non-negative on $[a,b]$, with zeros only at the endpoints. Analysis of the derivative of the integral function shows $x = b$ is a local extremum point.</p>
Correct Answer: A