Question:

Step 1: Identify the optimization problem and apply weighted AM-GM inequality. We need to maximize $a^2b^3c^4$ subject to the constraint $a + b + c = 9$ where $a, b, c \geq 0$. We rewrite the constraint by grouping terms strategically: $$a + b + c = \underbrace{\frac{a}{2} + \frac{a}{2}}_{2 \text{ terms}} + \underbrace{\frac{b}{3} + \frac{b}{3} + \frac{b}{3}}_{3 \text{ terms}} + \underbrace{\frac{c}{4} + \frac{c}{4} + \frac{c}{4} + \frac{c}{4}}_{4 \text{ terms}} = 9$$ Step 2: Apply the weighted AM-GM inequality. By the AM-GM inequality applied to these 9 terms: $$\frac{\frac{a}{2} + \frac{a}{2} + \frac{b}{3} + \frac{b}{3} + \frac{b}{3} + \frac{c}{4} + \frac{c}{4} + \frac{c}{4} + \frac{c}{4}}{9} \geq \sqrt[9]{\left(\frac{a}{2}\right)^2 \left(\frac{b}{3}\right)^3 \left(\frac{c}{4}\right)^4}$$ Simplifying the left side: $$\frac{9}{9} = 1 \geq \sqrt[9]{\left(\frac{a}{2}\right)^2 \left(\frac{b}{3}\right)^3 \left(\frac{c}{4}\right)^4}$$ Step 3: Determine when equality holds in AM-GM. Equality in AM-GM occurs when all terms are equal: $$\frac{a}{2} = \frac{b}{3} = \frac{c}{4} = k$$ This gives us: $$a = 2k, \quad b = 3k, \quad c = 4k$$ Step 4: Use the constraint to find the value of $k$. Substituting into $a + b + c = 9$: $$2k + 3k + 4k = 9$$ $$9k = 9$$ $$k = 1$$ Therefore: $a = 2$, $b = 3$, $c = 4$ Step 5: Calculate the maximum value of $a^2b^3c^4$. $$a^2b^3c^4 = 2^2 \cdot 3^3 \cdot 4^4$$ Express $4^4$ in terms of prime factors: $$4^4 = (2^2)^4 = 2^8$$ Therefore: $$a^2b^3c^4 = 2^2 \cdot 3^3 \cdot 2^8 = 2^{10} \cdot 3^3$$ Step 6: Identify the values of $x$ and $y$. From the expression $2^{10} \cdot 3^3$, we have: $$x = 10, \quad y = 3$$ Step 7: Calculate $\log_{10}(x^y)$. $$\log_{10}(x^y) = \log_{10}(10^3) = 3\log_{10}(10) = 3 \cdot 1 = 3$$ The value of $\log_{10}(x^y) = 3$, which corresponds to **Option 2**.