If `xyz=abc,` then the least value of `bcx+cay+abz,` is

Options

A. 3abc
B. 6abc
C. abc
D. 4abc

Answer

To find the least value of \(bcx + cay + abz\) given that \(xyz = abc\), we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Identify the Expression**: We need to minimize the expression \(S = bcx + cay + abz\).

2. **Apply AM-GM Inequality**: According to the AM-GM inequality, for any non-negative real numbers \(u, v, w\):
\[ \frac{u + v + w}{3} \geq \sqrt[3]{uvw} \]
We can apply this to our terms \(bcx\), \(cay\), and \(abz\).

3. **Set Up the Inequality**:
\[ \frac{bcx + cay + abz}{3} \geq \sqrt[3]{(bcx)(cay)(abz)} \]

4. **Calculate the Product**:
\[ (bcx)(cay)(abz) = b^2c^2a^2xyz \]
Since we know \(xyz = abc\), we can substitute:
\[ (bcx)(cay)(abz) = b^2c^2a^2(abc) = a^3b^3c^3 \]

5. **Substitute Back into the Inequality**:
\[ \frac{bcx + cay + abz}{3} \geq \sqrt[3]{a^3b^3c^3} = abc \]

6. **Multiply Both Sides by 3**:
\[ bcx + cay + abz \geq 3abc \]

7. **Conclusion**: The least value of \(bcx + cay + abz\) is \(3abc\).

### Final Answer:
The least value of \(bcx + cay + abz\) is \(3abc\).

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