Let a, b, c be positive numbers, then the minimum value of `(a^4+b^4+c^2)/(abc)`

Options

A. 4
B. `2 ^(3//4)`
C. ` sqrt2`
D. `2 sqrt2`

Answer

To find the minimum value of the expression \(\frac{a^4 + b^4 + c^2}{abc}\) for positive numbers \(a\), \(b\), and \(c\), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We want to minimize the expression:
\[ \frac{a^4 + b^4 + c^2}{abc} \]

2. **Apply AM-GM Inequality**:
We can apply the AM-GM inequality to the terms \(a^4\), \(b^4\), \(\frac{c^2}{2}\), and \(\frac{c^2}{2}\). According to the AM-GM inequality:
\[ \frac{a^4 + b^4 + \frac{c^2}{2} + \frac{c^2}{2}}{4} \geq \sqrt[4]{a^4 \cdot b^4 \cdot \frac{c^2}{2} \cdot \frac{c^2}{2}} \]

3. **Simplify the Right Side**:
The right side simplifies to:
\[ \sqrt[4]{a^4 \cdot b^4 \cdot \frac{c^4}{4}} = \sqrt[4]{\frac{a^4 b^4 c^4}{4}} = \frac{(abc)^4}{4^{1/4}} = \frac{abc}{2} \]

4. **Combine the Inequalities**:
From the AM-GM inequality, we have:
\[ a^4 + b^4 + c^2 \geq 4 \cdot \frac{abc}{2} = 2abc \]

5. **Substitute Back into the Original Expression**:
Now substituting back into our original expression:
\[ \frac{a^4 + b^4 + c^2}{abc} \geq \frac{2abc}{abc} = 2 \]

6. **Find the Minimum Value**:
To find the minimum value, we need to check if equality holds in the AM-GM inequality. Equality holds when:
\[ a^4 = b^4 = \frac{c^2}{2} = k \quad \text{for some } k \]
This implies:
\[ a = b = \sqrt[4]{k}, \quad c = \sqrt{2k} \]

7. **Calculate the Minimum Value**:
Substituting \(k = 1\) gives:
\[ a = b = 1, \quad c = \sqrt{2} \]
Thus:
\[ \frac{a^4 + b^4 + c^2}{abc} = \frac{1^4 + 1^4 + (\sqrt{2})^2}{1 \cdot 1 \cdot \sqrt{2}} = \frac{1 + 1 + 2}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

### Conclusion:
The minimum value of \(\frac{a^4 + b^4 + c^2}{abc}\) is \(2\sqrt{2}\).