Ifx+ y + z =13,then the maximum value of(x -2)(y+ 1) (z - 3) is

Options

A. 25
B. 30
C. 54
D. 27

Answer

To find the maximum value of the expression \((x - 2)(y + 1)(z - 3)\) given the constraint \(x + y + z = 13\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (x - 2)(y + 1)(z - 3) \]

### Step 2: Substitute variables
Let:
\[ a = x - 2, \quad b = y + 1, \quad c = z - 3 \]
Then we can express \(x\), \(y\), and \(z\) in terms of \(a\), \(b\), and \(c\):
\[ x = a + 2, \quad y = b - 1, \quad z = c + 3 \]

### Step 3: Substitute into the constraint
Substituting these into the constraint \(x + y + z = 13\):
\[ (a + 2) + (b - 1) + (c + 3) = 13 \]
This simplifies to:
\[ a + b + c + 4 = 13 \]
Thus:
\[ a + b + c = 9 \]

### Step 4: Find the maximum product
We want to maximize the product:
\[ abc \]
Given the constraint \(a + b + c = 9\). By the AM-GM inequality, the product \(abc\) is maximized when \(a = b = c\).

### Step 5: Set \(a\), \(b\), and \(c\) equal
Let:
\[ a = b = c = k \]
Then:
\[ 3k = 9 \implies k = 3 \]

### Step 6: Calculate the maximum value
Now substituting back:
\[ a = 3, \quad b = 3, \quad c = 3 \]
The maximum product is:
\[ abc = 3 \times 3 \times 3 = 27 \]

### Conclusion
Thus, the maximum value of \((x - 2)(y + 1)(z - 3)\) is:
\[ \boxed{27} \]

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