If `a, b, c in R^(+)` such that `a+b+c=27`, then the maximum value of `a^(2)b^(3)c^(4)` is equal to

Options

A. `2^(8). 3^(10)`
B. `2^(9).3^(12)`
C. `2^(10).3^(12)`
D. `2^(11).3^(13)`

Answer

To find the maximum value of \( a^2 b^3 c^4 \) given the constraint \( a + b + c = 27 \) and \( a, b, c \in \mathbb{R}^+ \), we can use the method of Lagrange multipliers or apply the AM-GM inequality. Here, we will use the AM-GM inequality.

### Step-by-Step Solution:

1. **Set Up the Problem**:
We want to maximize the expression \( a^2 b^3 c^4 \) under the constraint \( a + b + c = 27 \).

2. **Apply the AM-GM Inequality**:
According to the AM-GM inequality, for any non-negative real numbers \( x_1, x_2, \ldots, x_n \):
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \ldots x_n} \]
We can apply this to the terms \( a, a, b, b, b, c, c, c, c \) (where \( a \) appears twice, \( b \) three times, and \( c \) four times).

3. **Calculate the Arithmetic Mean**:
The total number of terms is \( 2 + 3 + 4 = 9 \). Therefore, we have:
\[ \frac{a + a + b + b + b + c + c + c + c}{9} = \frac{2a + 3b + 4c}{9} \]
By the AM-GM inequality:
\[ \frac{2a + 3b + 4c}{9} \geq \sqrt[9]{a^2 b^3 c^4} \]

4. **Substituting the Constraint**:
Since \( a + b + c = 27 \), we can express \( 2a + 3b + 4c \) in terms of \( a + b + c \):
\[ 2a + 3b + 4c = 2a + 3b + 4(27 - a - b) = 2a + 3b + 108 - 4a - 4b = -2a - b + 108 \]
Thus, we have:
\[ \frac{-2a - b + 108}{9} \geq \sqrt[9]{a^2 b^3 c^4} \]

5. **Finding the Maximum Value**:
To maximize \( a^2 b^3 c^4 \), we need to find the equality condition in the AM-GM inequality, which occurs when:
\[ a = a = b = b = b = c = c = c = c \]
Let \( a = 2k \), \( b = 3k \), and \( c = 4k \). Then:
\[ 2k + 3k + 4k = 27 \implies 9k = 27 \implies k = 3 \]
Therefore:
\[ a = 6, \quad b = 9, \quad c = 12 \]

6. **Calculate the Maximum Value**:
Now substitute \( a, b, c \) back into the expression:
\[ a^2 b^3 c^4 = 6^2 \cdot 9^3 \cdot 12^4 \]
Calculate each term:
\[ 6^2 = 36, \quad 9^3 = 729, \quad 12^4 = 20736 \]
Now multiply:
\[ 36 \cdot 729 \cdot 20736 \]
Calculate step by step:
\[ 36 \cdot 729 = 26244 \]
Then:
\[ 26244 \cdot 20736 = 544864320 \]

Thus, the maximum value of \( a^2 b^3 c^4 \) is \( 544864320 \).