If `x + y + z =21`, then the maximum value of `(x - 6) (y + 7)(z - 4)` is :

Options

A. 343
B. 216
C. 125
D. not unique

Answer

To find the maximum value of the expression \((x - 6)(y + 7)(z - 4)\) given the constraint \(x + y + z = 21\), we can use the method of inequalities, specifically the AM-GM inequality.

### Step-by-Step Solution

1. **Rearranging the Expression**:
We start with the expression \((x - 6)(y + 7)(z - 4)\). To simplify our calculations, we can express \(x\), \(y\), and \(z\) in terms of a new variable:
- Let \(a = x - 6\)
- Let \(b = y + 7\)
- Let \(c = z - 4\)

Then, we can rewrite the original equation:
\[ x + y + z = 21 \implies (a + 6) + (b - 7) + (c + 4) = 21 \]
Simplifying this gives:
\[ a + b + c + 3 = 21 \implies a + b + c = 18 \]

2. **Applying AM-GM Inequality**:
According to the AM-GM inequality, for non-negative numbers \(a\), \(b\), and \(c\):
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
Substituting \(a + b + c = 18\):
\[ \frac{18}{3} \geq \sqrt[3]{abc} \implies 6 \geq \sqrt[3]{abc} \]

3. **Cubing Both Sides**:
To eliminate the cube root, we cube both sides:
\[ 6^3 \geq abc \implies 216 \geq abc \]
This tells us that the maximum value of \((x - 6)(y + 7)(z - 4)\) is at most 216.

4. **Finding When Equality Holds**:
The equality in the AM-GM inequality holds when \(a = b = c\). Thus, we set:
\[ a = b = c = 6 \]
Therefore:
- \(x - 6 = 6 \implies x = 12\)
- \(y + 7 = 6 \implies y = -1\)
- \(z - 4 = 6 \implies z = 10\)

Checking the sum:
\[ x + y + z = 12 + (-1) + 10 = 21 \]
This satisfies the original condition.

5. **Conclusion**:
The maximum value of \((x - 6)(y + 7)(z - 4)\) is:
\[ (6)(6)(6) = 216 \]

### Final Answer:
The maximum value of \((x - 6)(y + 7)(z - 4)\) is **216**.