<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>
Step-by-Step Solution
Key Concept: Use the constraint f(x) + g(x) + h(x) = 2 to expand (f + g + h)² = 4, then apply Cauchy-Schwarz or explore specific function relationships to determine bounds on the integral of sum of squares.
<p><strong>Step 1:</strong> Given constraint: f(x) + g(x) + h(x) = 2 for all x ∈ ℝ</p><p><strong>Step 2:</strong> Square the constraint: [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> By Cauchy-Schwarz inequality, the cross terms are minimized when f = g = h = 2/3, giving f² + g² + h² ≥ 4/3 at each point.</p><p><strong>Step 6:</strong> Integrate from 0 to 3/4: ∫₀^(3/4) (f² + g² + h²)dx ≥ (4/3)(3/4) = 1</p><p><strong>Step 7:</strong> The minimum value is achieved when f(x) = g(x) = h(x) = 2/3, making the integral equal to exactly 1.</p><p>∴ Answer: C (The value equals 1, or lies in a specific range with minimum value 1)</p>
Correct Answer: C