Let $x$, $y$ be positive real numbers such that $xy^3 = 81$, then:
Step-by-Step Solution
Key Concept: Apply AM-GM inequality by splitting the linear expression into four parts whose product involves $xy^3$, the given constraint. The split must be chosen so that the geometric mean simplifies using $xy^3=81$.
Step 1:
To solve this problem, we first need to understand the given condition, which is $xy^3 = 81$, where $x$ and $y$ are positive real numbers. We are asked to minimize the expressions $(x+y)^4$, $(x+3y)^4$, $(3x+y)^4$, and $(2x+3y)^4$ using the AM-GM inequality.
Step 2:
Let's start by minimizing the expression $(x+y)^4$. We can rewrite $x + y$ as $x + \frac{y}{3} + \frac{y}{3} + \frac{y}{3}$. By applying the AM-GM inequality to these four terms, we get:
$$\frac{x + \frac{y}{3} + \frac{y}{3} + \frac{y}{3}}{4} \geq \left(x \cdot \frac{y}{3} \cdot \frac{y}{3} \cdot \frac{y}{3}\right)^{1/4} = \left(\frac{xy^3}{27}\right)^{1/4} = \left(\frac{81}{27}\right)^{1/4} = 3^{1/4}.$$
This implies that $x+y \geq 4 \cdot 3^{1/4}$, and therefore, $(x+y)^4 \geq 4^4 \cdot 3 = 256 \cdot 3 = 768 = 3 \cdot 2^8$.
Step 3:
Next, we minimize the expression $(x+3y)^4$. We can express $x+3y$ as $x + y + y + y$. Applying the AM-GM inequality, we have:
$$\frac{x+y+y+y}{4} \geq (xy^3)^{1/4} = 81^{1/4} = 3.$$
This gives us $x+3y \geq 12$, and consequently, $(x+3y)^4 \geq 12^4$.
Step 4:
Now, let's minimize the expression $(3x+y)^4$. We rewrite $3x+y$ as $3x + \frac{y}{3}+\frac{y}{3}+\frac{y}{3}$. Using the AM-GM inequality, we obtain:
$$\frac{3x+\frac{y}{3}+\frac{y}{3}+\frac{y}{3}}{4} \geq \left(3x\cdot\frac{y}{3}\cdot\frac{y}{3}\cdot\frac{y}{3}\right)^{1/4} = \left(\frac{xy^3}{9}\right)^{1/4} = \left(\frac{81}{9}\right)^{1/4} = 9^{1/4}.$$
This implies that $3x+y \geq 4\cdot 9^{1/4}$, and hence, $(3x+y)^4 \geq 256\cdot 9 = 9(2)^8$.
Step 5:
Finally, we minimize the expression $(2x+3y)^4$. We express $2x+3y$ as $2x + y + y + y$. Applying the AM-GM inequality, we get:
$$\frac{2x+y+y+y}{4} \geq (2x\cdot y^3)^{1/4} = (2\cdot 81)^{1/4} = (162)^{1/4}.$$
This yields $(2x+3y)^4 \geq 256\cdot 162 = 41472$. Notably, $2(12)^4 = 2\cdot 20736 = 41472$, which matches our result for $(2x+3y)^4$.
Step 6:
Based on our calculations, we can conclude that the correct answer is the option that corresponds to the minimum value of $(2x+3y)^4$, which is option 1. Therefore, the final answer is $\boxed{1}$.
Correct Answer: 1