If `a,b,c` are positive real numbers and `2a+b+3c=1`, then the maximum value of `a^(4)b^(2)c^(2)` is equal to

Options

A. `(1)/(3*4^(8))`
B. `(1)/(9*4^(7))`
C. `(1)/(9*4^(8))`
D. `(1)/(27*4^(8))`

Explanation

To find the maximum value of \( a^4 b^2 c^2 \) given the constraint \( 2a + b + 3c = 1 \) and \( a, b, c > 0 \), we can use the method of Lagrange multipliers or apply the AM-GM inequality.

### Step 1: Apply the AM-GM Inequality

We can express the constraint in a way that allows us to apply the AM-GM inequality. We want to maximize \( a^4 b^2 c^2 \).

We can rewrite the constraint as:
\[ 2a + b + 3c = 1 \]

We will distribute the terms in a way that fits the AM-GM inequality. We can express \( 2a \) as \( a + a \) and \( 3c \) as \( c + c + c \). Thus, we can rewrite the expression as:
\[ a + a + b + c + c + c = 1 \]

### Step 2: Apply AM-GM

Now, we apply the AM-GM inequality to the six terms \( a, a, b, c, c, c \):
\[ \frac{a + a + b + c + c + c}{6} \geq \sqrt[6]{a^2 b c^3} \]

This simplifies to:
\[ \frac{1}{6} \geq \sqrt[6]{a^2 b c^3} \]

### Step 3: Raise Both Sides to the Power of 6

Raising both sides to the power of 6 gives:
\[ \left(\frac{1}{6}\right)^6 \geq a^2 b c^3 \]

### Step 4: Relate to the Desired Expression

We want to find \( a^4 b^2 c^2 \). Notice that:
\[ a^4 b^2 c^2 = (a^2 b c^3) \cdot (a^2 b^{-1} c^{-1}) \cdot c^2 \]

To maximize \( a^4 b^2 c^2 \), we can express it in terms of \( a^2 b c^3 \):
\[ a^4 b^2 c^2 = (a^2 b c^3) \cdot (a^2 b^{-1} c^{-1}) \cdot c^2 \]

### Step 5: Find Maximum Value

Using the AM-GM inequality again, we can find the maximum value of \( a^4 b^2 c^2 \) under the constraint.

Let’s denote:
\[ x_1 = a, \quad x_2 = a, \quad x_3 = b, \quad x_4 = c, \quad x_5 = c, \quad x_6 = c \]

Then, we have:
\[ \frac{2a + b + 3c}{6} \geq \sqrt[6]{a^2 b c^3} \]

Substituting \( 2a + b + 3c = 1 \):
\[ \frac{1}{6} \geq \sqrt[6]{a^2 b c^3} \]

### Step 6: Calculate the Maximum Value

Now, we can find the maximum value of \( a^4 b^2 c^2 \):
\[ \left(\frac{1}{6}\right)^6 = \frac{1}{6^6} = \frac{1}{46656} \]

Thus, the maximum value of \( a^4 b^2 c^2 \) is:
\[ \text{Maximum value} = \frac{1}{6^6} \cdot \text{(some constant)} \]

To find the exact maximum value, we can use the equality condition of AM-GM, which occurs when all the terms are equal:
\[ a = a = b = c = \frac{1}{6} \]

### Final Answer

The maximum value of \( a^4 b^2 c^2 \) is:
\[ \frac{1}{6^6} = \frac{1}{46656} \]