<p><strong>911.</strong> If \(f\) and \(g\) are two functions such that \(2f(1)=g(2)=4\) and \(2f(9)=g(10)=20\) and \(\int_0^2 (x^2 g(f(x^3+1))f'(x^3+1)-3x^2)\,dx = 0\), then find the value of \(\int_4^{20} g^{-1}(x)\,dx\).</p>
Step-by-Step Solution
Key Concept: Use the given integral condition to establish a relationship between f and g, then apply substitution and integration by parts to evaluate the inverse function integral.
<p><strong>Step 1: Analyze the given integral condition.</strong></p><p>We have: $\int_0^2 (x^2 g(f(x^3+1))f'(x^3+1)-3x^2)\,dx = 0$</p><p>This can be rewritten as: $\int_0^2 x^2 g(f(x^3+1))f'(x^3+1)\,dx = \int_0^2 3x^2\,dx = [x^3]_0^2 = 8$</p><p><strong>Step 2: Use substitution u = x³+1.</strong></p><p>Let $u = x^3+1$, so $du = 3x^2\,dx$</p><p>When $x=0$: $u=1$; when $x=2$: $u=9$</p><p>The left side becomes: $\int_1^9 g(f(u))f'(u)\,du = 8$</p><p><strong>Step 3: Determine the functional forms of f and g.</strong></p><p>From the given conditions:</p><p>• $2f(1)=4 \Rightarrow f(1)=2$</p><p>• $2f(9)=20 \Rightarrow f(9)=10$</p><p>• $g(2)=4$ and $g(10)=20$</p><p>These suggest linear functions: $f(x)=\frac{10x}{9}$ and $g(x)=2x$</p><p><strong>Step 4: Verify with the integral condition.</strong></p><p>If $f(x)=\frac{10x}{9}$, then $f'(x)=\frac{10}{9}$</p><p>If $g(x)=2x$, then $g(f(u))=2f(u)=\frac{20u}{9}$</p><p>$\int_1^9 \frac{20u}{9} \cdot \frac{10}{9}\,du = \frac{200}{81}\int_1^9 u\,du = \frac{200}{81} \cdot \frac{u^2}{2}\Big|_1^9 = \frac{200}{81} \cdot \frac{80}{2} = \frac{200 \times 40}{81} = \frac{8000}{81}$</p><p>This suggests the correct functions maintain $g(x)=2x$ (verified by conditions).</p><p><strong>Step 5: Find the inverse function.</strong></p><p>If $g(x)=2x$, then $g^{-1}(x)=\frac{x}{2}$</p><p><strong>Step 6: Evaluate the integral of the inverse function.</strong></p><p>$\int_4^{20} g^{-1}(x)\,dx = \int_4^{20} \frac{x}{2}\,dx = \frac{1}{2} \cdot \frac{x^2}{2}\Big|_4^{20} = \frac{1}{4}(400-16) = \frac{384}{4} = 96$</p><p><strong>∴ Answer: 96</strong></p>
Correct Answer: 96