Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>If \(f(x) + g(x) + h(x) = 2\) \(\forall\, x \in R\), then the value of the expression \(\displaystyle\int_0^{3/4} \left(f^2(x) + g^2(x) + h^2(x)\right) dx\) can be:</p>
<p>\(\dfrac{1}{2}\)</p>
<p>\(1\)</p>
<p>\(\dfrac{3}{2}\)</p>
<p>\(4\)</p>
Step-by-Step Solution
Key Concept: Use the constraint f(x) + g(x) + h(x) = 2 to expand (f + g + h)² = 4, then integrate to relate the sum of squares integral to cross-product terms, which can be minimized or bounded using Cauchy-Schwarz inequality.
<p><strong>Step 1:</strong> Use the given constraint: f(x) + g(x) + h(x) = 2 for all x ∈ ℝ</p><p><strong>Step 2:</strong> Square both sides: [f(x) + g(x) + h(x)]² = 4</p><p><strong>Step 3:</strong> Expand: f²(x) + g²(x) + h²(x) + 2[f(x)g(x) + g(x)h(x) + h(x)f(x)] = 4</p><p><strong>Step 4:</strong> Rearrange: f²(x) + g²(x) + h²(x) = 4 - 2[f(x)g(x) + g(x)h(x) + h(x)f(x)]</p><p><strong>Step 5:</strong> Integrate from 0 to 3/4:</p><p>∫₀^(3/4) [f²(x) + g²(x) + h²(x)]dx = 4(3/4) - 2∫₀^(3/4) [f(x)g(x) + g(x)h(x) + h(x)f(x)]dx</p><p><strong>Step 6:</strong> The integral equals 3 - 2∫₀^(3/4) [f(x)g(x) + g(x)h(x) + h(x)f(x)]dx</p><p><strong>Step 7:</strong> By Cauchy-Schwarz, the cross terms are bounded. The minimum value (when f = g = h = 2/3) gives: ∫₀^(3/4) (3 · 4/9)dx = 3 · 4/9 · 3/4 = 1, and maximum bounds depend on cross-product behavior.</p><p>∴ Answer: B</p>
Correct Answer: B