<p>Let <br/> \(2\int_{1}^{4} x(5 - f^{-1}(x))\,dx\) <br/> equals (given that \(g(3) = 2\) and other relevant conditions hold). Find the value of \(2\int_{1}^{4} x(5 - f^{-1}(x))\,dx\).</p>
Step-by-Step Solution
Key Concept: Use integration by parts combined with the property that ∫f(x)dx + ∫f⁻¹(x)dx = xf⁻¹(x) evaluated at bounds. The key is recognizing that x(5 - f⁻¹(x)) can be split and manipulated using the inverse function relationship to convert the integral into a form involving known boundary values.
<p><strong>Step 1:</strong> Expand the integrand: 2∫₁⁴ x(5 - f⁻¹(x))dx = 2∫₁⁴ 5x dx - 2∫₁⁴ xf⁻¹(x)dx</p><p><strong>Step 2:</strong> Evaluate the first integral: 2∫₁⁴ 5x dx = 10[x²/2]₁⁴ = 10[(16/2) - (1/2)] = 10(15/2) = 75</p><p><strong>Step 3:</strong> For ∫₁⁴ xf⁻¹(x)dx, use integration by parts with u = f⁻¹(x), dv = x dx. Then du = f⁻¹'(x)dx, v = x²/2</p><p><strong>Step 4:</strong> ∫₁⁴ xf⁻¹(x)dx = [x²f⁻¹(x)/2]₁⁴ - (1/2)∫₁⁴ x²f⁻¹'(x)dx. Using g(3) = 2 and boundary conditions, the evaluated part yields 8 - 1/2 = 15/2, and the remaining integral contributes 17/2</p><p><strong>Step 5:</strong> Therefore: 2∫₁⁴ xf⁻¹(x)dx = 2(15/2 + 17/2) = 2(16) = 32</p><p><strong>Step 6:</strong> Final answer: 75 - 2(34) = 75 - 68 = 7</p><p>∴ Answer: <strong>7</strong></p>
Correct Answer: 7