If a,b,c are positive real numbers such that a + b +c=18, find the maximum value of `a^2b^3c^4`

Options

A. `2^(19)xx3^(2)`
B. `2^(19)xx3^(3)`
C. `2^(18)xx3^(3)`
D. none of these

Explanation

To find the maximum value of \( a^2b^3c^4 \) given that \( a + b + c = 18 \) and \( a, b, c \) are positive real numbers, we can use the method of inequalities, specifically the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Set Up the Problem**: We know that \( a + b + c = 18 \). We want to maximize the expression \( a^2b^3c^4 \).

2. **Apply AM-GM Inequality**:
The AM-GM inequality states that for any non-negative numbers \( x_1, x_2, \ldots, x_n \):
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \ldots x_n} \]
In our case, we can express \( a, b, c \) in terms of their contributions to the product \( a^2b^3c^4 \). We will divide \( a \) into 2 parts, \( b \) into 3 parts, and \( c \) into 4 parts.

3. **Rewrite the Terms**:
We can rewrite the sum \( a + b + c \) as:
\[ \frac{a/2 + a/2 + b/3 + b/3 + b/3 + c/4 + c/4 + c/4 + c/4}{9} \geq \sqrt[9]{\left(\frac{a}{2}\right)^2 \left(\frac{b}{3}\right)^3 \left(\frac{c}{4}\right)^4} \]

4. **Calculate the Left Side**:
Since \( a + b + c = 18 \), we have:
\[ \frac{18}{9} = 2 \]

5. **Set Up the Inequality**:
Thus, we can write:
\[ 2 \geq \sqrt[9]{\left(\frac{a}{2}\right)^2 \left(\frac{b}{3}\right)^3 \left(\frac{c}{4}\right)^4} \]

6. **Raise Both Sides to the Power of 9**:
\[ 2^9 \geq \left(\frac{a}{2}\right)^2 \left(\frac{b}{3}\right)^3 \left(\frac{c}{4}\right)^4 \]

7. **Simplify the Right Side**:
\[ 512 \geq \frac{a^2 b^3 c^4}{2^2 \cdot 3^3 \cdot 4^4} \]

8. **Calculate the Denominator**:
\[ 2^2 = 4, \quad 3^3 = 27, \quad 4^4 = 256 \]
Thus,
\[ 2^2 \cdot 3^3 \cdot 4^4 = 4 \cdot 27 \cdot 256 = 27648 \]

9. **Final Inequality**:
\[ 512 \cdot 27648 \geq a^2 b^3 c^4 \]

10. **Calculate the Maximum Value**:
\[ a^2 b^3 c^4 \leq 512 \cdot 27648 \]

11. **Compute the Product**:
\[ 512 \cdot 27648 = 14155776 \]

Thus, the maximum value of \( a^2b^3c^4 \) given the constraint \( a + b + c = 18 \) is \( 14155776 \).