Definite Integration
Grade 12
Question:
<p>If \(f\) is positive and decreasing on \([a,b]\) and \(g\) is integrable, then \(\exists\,c\in[a,b]\) such that \(\int_a^b f(x)g(x)\,dx =\) ? [JEE Advanced 2011]</p>
f(a) \cdot \intₐ^c g(x)dx
f(b) \cdot \intₐ^b g(x)dx
f(c) \cdot \intₐ^b g(x)dx
\intₐ^c f(x)dx
Step-by-Step Solution
Key Concept: Bonnet's MVT: if f decreasing, positive, then \intₐ^b fg = f(a)\intₐ^c g for some c\in [a,b].
<div class='solution'>
<p><strong>Bonnet's First Mean Value Theorem:</strong> If $f$ is positive and decreasing on $[a,b]$, then $\exists\,c\in[a,b]$:</p>
<p>$$\int_a^b f(x)g(x)\,dx = f(a)\int_a^c g(x)\,dx$$</p>
<p>This is distinct from the regular MVT ($f(c)\int_a^b g$) because $f$ is not assumed constant.</p>
Correct Answer: A