If A, B, C be an acute angled triangle, then the minimum value of `tan^(4)A+tan^(4)B+tan^(4)C` will be

Options

A. 729
B. 27
C. `81sqrt3`
D. none of these

Explanation

To find the minimum value of \( \tan^4 A + \tan^4 B + \tan^4 C \) for an acute-angled triangle \( ABC \), we can use the properties of symmetric functions and inequalities. Here’s a step-by-step solution:

### Step 1: Use the AM-GM Inequality
We can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality. For three non-negative numbers \( x_1, x_2, x_3 \):
\[ \frac{x_1 + x_2 + x_3}{3} \geq \sqrt[3]{x_1 x_2 x_3} \]
In our case, let \( x_1 = \tan^4 A \), \( x_2 = \tan^4 B \), and \( x_3 = \tan^4 C \).

### Step 2: Set Up the Inequality
Applying AM-GM gives us:
\[ \frac{\tan^4 A + \tan^4 B + \tan^4 C}{3} \geq \sqrt[3]{\tan^4 A \tan^4 B \tan^4 C} \]
This implies:
\[ \tan^4 A + \tan^4 B + \tan^4 C \geq 3 \sqrt[3]{\tan^4 A \tan^4 B \tan^4 C} \]

### Step 3: Use the Condition for Angles in a Triangle
Since \( A + B + C = 180^\circ \) and \( A, B, C \) are acute angles, we can use the fact that \( \tan A + \tan B + \tan C \) is minimized when \( A = B = C = 60^\circ \).

### Step 4: Calculate \( \tan 60^\circ \)
We know:
\[ \tan 60^\circ = \sqrt{3} \]
Thus:
\[ \tan^4 60^\circ = (\sqrt{3})^4 = 9 \]

### Step 5: Substitute Back into the Expression
If \( A = B = C = 60^\circ \):
\[ \tan^4 A + \tan^4 B + \tan^4 C = 9 + 9 + 9 = 27 \]

### Conclusion
Therefore, the minimum value of \( \tan^4 A + \tan^4 B + \tan^4 C \) is:
\[ \boxed{27} \]