Definite Integration
Integral Inequalities
Grade None
Question:
<p>Let \(f,g\) be continuous on \([a,b]\) with \(f(x)\le g(x)\). Which are always true?</p>
\intₐ^b f(x)dx \leq \intₐ^b g(x)dx
|\intₐ^b f(x)dx| \leq \intₐ^b |f(x)|dx
(\intₐ^b f \cdot g dx)^2 = \intₐ^b f^2dx \cdot \intₐ^b g^2dx
(\intₐ^b f \cdot g dx)^2 \leq \intₐ^b f^2dx \cdot \intₐ^b g^2dx
Step-by-Step Solution
Key Concept: A: monotonicity. B: triangle inequality for integrals. C: equality only when f\proptog — generally false. D: Cauchy-Schwarz inequality for integrals.
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<p><strong>A:</strong> ✓ Monotonicity: \(f\le g\Rightarrow\int f\le\int g\).</p>
<p><strong>B:</strong> ✓ \(\left|\int_a^b f\right|\le\int_a^b|f|\) — the integral triangle inequality.</p>
<p><strong>C:</strong> ✗ Equality in Cauchy-Schwarz holds iff \(f=\lambda g\) for some constant \(\lambda\). Not always equal.</p>
<p><strong>D:</strong> ✓ Cauchy-Schwarz inequality for integrals: \(\left(\int fg\right)^2\le\int f^2\cdot\int g^2\).</p>
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Correct Answer: ['A', 'B', 'D']