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 \(\frac{a^4 + b^4 + c^2}{abc}\), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality. Here’s a step-by-step solution:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \frac{a^4 + b^4 + c^2}{abc} \]
We can rewrite \(c^2\) as:
\[ c^2 = \frac{c^2}{2} + \frac{c^2}{2} \]
This allows us to apply the AM-GM inequality effectively.

### Step 2: Apply AM-GM Inequality
Now, we apply the AM-GM inequality to the terms \(a^4\), \(b^4\), \(\frac{c^2}{2}\), and \(\frac{c^2}{2}\):
\[ \frac{a^4 + b^4 + \frac{c^2}{2} + \frac{c^2}{2}}{4} \geq \sqrt[4]{a^4 \cdot b^4 \cdot \frac{c^2}{2} \cdot \frac{c^2}{2}} \]
This simplifies to:
\[ \frac{a^4 + b^4 + c^2}{4} \geq \sqrt[4]{a^4 b^4 \cdot \frac{c^4}{4}} = \sqrt[4]{\frac{a^4 b^4 c^4}{4}} \]

### Step 3: Simplify the Right Side
The right side can be simplified further:
\[ \sqrt[4]{\frac{(abc)^4}{4}} = \frac{abc}{\sqrt[4]{4}} = \frac{abc}{2^{1/2}} = \frac{abc}{\sqrt{2}} \]

### Step 4: Multiply Both Sides by 4
Multiplying both sides of the inequality by 4 gives:
\[ a^4 + b^4 + c^2 \geq \frac{4abc}{\sqrt{2}} \]

### Step 5: Divide by \(abc\)
Now, we divide both sides by \(abc\):
\[ \frac{a^4 + b^4 + c^2}{abc} \geq \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

### Step 6: Conclusion
Thus, the minimum value of \(\frac{a^4 + b^4 + c^2}{abc}\) is \(2\sqrt{2}\).