If a + b + c = 13 then, what is the maximum value of (a - 3) (b - 2) (c + 1)

Options

A. 27
B. 28
C. 32
D. 64

Answer

To solve the problem of finding the maximum value of the expression \((a - 3)(b - 2)(c + 1)\) given that \(a + b + c = 13\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression \((a - 3)(b - 2)(c + 1)\). To make it easier to work with, we can express \(a\), \(b\), and \(c\) in terms of a new variable \(k\):
- Let \(x = a - 3\)
- Let \(y = b - 2\)
- Let \(z = c + 1\)

From these definitions, we can express \(a\), \(b\), and \(c\) as:
- \(a = x + 3\)
- \(b = y + 2\)
- \(c = z - 1\)

### Step 2: Substitute into the sum equation
Now, substituting these values into the equation \(a + b + c = 13\):
\[ (x + 3) + (y + 2) + (z - 1) = 13 \]
This simplifies to:
\[ x + y + z + 4 = 13 \]
Thus, we have:
\[ x + y + z = 9 \]

### Step 3: Apply the AM-GM Inequality
To find the maximum value of the product \(xyz\), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality. According to AM-GM:
\[ \frac{x + y + z}{3} \geq \sqrt[3]{xyz} \]
Substituting \(x + y + z = 9\) into the inequality gives:
\[ \frac{9}{3} \geq \sqrt[3]{xyz} \]
This simplifies to:
\[ 3 \geq \sqrt[3]{xyz} \]

### Step 4: Cube both sides
Cubing both sides of the inequality results in:
\[ 27 \geq xyz \]
This means that the maximum value of \(xyz\) is \(27\).

### Step 5: Find conditions for equality
The equality in AM-GM holds when \(x = y = z\). Therefore, we set:
\[ x = y = z \]
Since \(x + y + z = 9\), we can write:
\[ 3x = 9 \implies x = 3 \]
Thus, \(y = 3\) and \(z = 3\) as well.

### Step 6: Substitute back to find \(a\), \(b\), and \(c\)
Now substituting back to find \(a\), \(b\), and \(c\):
- \(a = x + 3 = 3 + 3 = 6\)
- \(b = y + 2 = 3 + 2 = 5\)
- \(c = z - 1 = 3 - 1 = 2\)

### Conclusion
Finally, substituting these values into the original expression:
\[ (a - 3)(b - 2)(c + 1) = (6 - 3)(5 - 2)(2 + 1) = 3 \cdot 3 \cdot 3 = 27 \]
Thus, the maximum value of \((a - 3)(b - 2)(c + 1)\) is **27**.