The minimum value of `27^(cos3x)81^(sin3x)` is

Options

A. `1//243`
B. -5
C. `1//5`
D. `1//3`

Answer

To find the minimum value of the expression \(27^{\cos(3x)} \cdot 81^{\sin(3x)}\), we can follow these steps:

### Step 1: Simplify the expression
We start by rewriting the bases in terms of powers of 3:
\[ 27 = 3^3 \quad \text{and} \quad 81 = 3^4 \]
Thus, we can rewrite the expression as:
\[ 27^{\cos(3x)} \cdot 81^{\sin(3x)} = (3^3)^{\cos(3x)} \cdot (3^4)^{\sin(3x)} \]
This simplifies to:
\[ 3^{3\cos(3x)} \cdot 3^{4\sin(3x)} = 3^{3\cos(3x) + 4\sin(3x)} \]

### Step 2: Define the function to minimize
Let:
\[ y = 3\cos(3x) + 4\sin(3x) \]
We need to find the minimum value of \(3^y\).

### Step 3: Find the minimum value of \(y\)
To find the minimum value of \(y\), we can use the formula for the minimum of a linear combination of sine and cosine:
\[ a\cos(\theta) + b\sin(\theta) \text{ has a minimum value of } -\sqrt{a^2 + b^2} \]
In our case, \(a = 3\) and \(b = 4\). Therefore:
\[ \text{Minimum value of } y = -\sqrt{3^2 + 4^2} = -\sqrt{9 + 16} = -\sqrt{25} = -5 \]

### Step 4: Substitute the minimum value back
Now, substituting this minimum value back into the expression for \(3^y\):
\[ 3^{\text{minimum value of } y} = 3^{-5} = \frac{1}{3^5} = \frac{1}{243} \]

### Conclusion
Thus, the minimum value of \(27^{\cos(3x)} \cdot 81^{\sin(3x)}\) is:
\[ \frac{1}{243} \]