Definite Integration
Bonnet MVT
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].
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<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>
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Correct Answer: A