If `x` lies between `-5` and `11`, then the greatest value of `(11-x)^(3)(x+5)^(5)` is

Options

A. `6^(5)*10^(3)`
B. `6^(3)*10^(3)`
C. `6^(3)*10^(4)`
D. `6^(3)*10^(5)`

Explanation

To find the greatest value of the expression \( (11 - x)^3 (x + 5)^5 \) for \( x \) in the interval \((-5, 11)\), we can use the concept of the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Here’s a step-by-step solution:

### Step 1: Define the variables
Let:
- \( a = 11 - x \)
- \( b = x + 5 \)

### Step 2: Express the product
We want to maximize the expression:
\[ P = a^3 b^5 \]

### Step 3: Find the relationship between \( a \) and \( b \)
From the definitions:
- \( a + b = (11 - x) + (x + 5) = 16 \)

### Step 4: Apply the AM-GM inequality
According to the AM-GM inequality:
\[ \frac{3a + 5b}{8} \geq (a^3 b^5)^{\frac{1}{8}} \]
This implies:
\[ 3a + 5b \geq 8 \sqrt[8]{a^3 b^5} \]

### Step 5: Substitute \( a + b \)
Since \( a + b = 16 \), we can express \( 3a + 5b \) in terms of \( a + b \):
\[ 3a + 5b = 3a + 5(16 - a) = 3a + 80 - 5a = 80 - 2a \]

### Step 6: Find the maximum value
To maximize \( P \), we need to minimize \( 2a \). Since \( a \) can range from \( 0 \) (when \( x = 11 \)) to \( 16 \) (when \( x = -5 \)), we want to find the optimal values of \( a \) and \( b \).

### Step 7: Set up the equality condition for AM-GM
For equality in AM-GM to hold, we need:
\[ 3a = 5b \]

### Step 8: Solve for \( a \) and \( b \)
From \( 3a = 5b \), we can express \( b \) in terms of \( a \):
\[ b = \frac{3}{5}a \]
Substituting into \( a + b = 16 \):
\[ a + \frac{3}{5}a = 16 \implies \frac{8}{5}a = 16 \implies a = 10 \]
Then,
\[ b = 16 - a = 6 \]

### Step 9: Substitute back to find maximum product
Now substituting \( a = 10 \) and \( b = 6 \) back into the expression for \( P \):
\[ P = (10)^3 (6)^5 \]

### Step 10: Calculate the maximum value
Calculating \( P \):
\[ P = 1000 \times 7776 = 7776000 \]

Thus, the greatest value of \( (11 - x)^3 (x + 5)^5 \) is:
\[ \boxed{7776000} \]