If `5^x+(2sqrt(3))^(2x)geq1 3^x` then the solution set for

Options

A. `[2,oo)`
B. {2}
C. `(-oo,2]`
D. [0,2]

Explanation

To solve the inequality \( 5^x + (2\sqrt{3})^{2x} \geq 13^x \), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ 5^x + (2\sqrt{3})^{2x} \geq 13^x \]
We can rewrite \( (2\sqrt{3})^{2x} \) as \( 4 \cdot 3^x \):
\[ 5^x + 4 \cdot 3^x \geq 13^x \]

### Step 2: Divide by \( 13^x \)
Next, we divide the entire inequality by \( 13^x \) (assuming \( 13^x > 0 \)):
\[ \frac{5^x}{13^x} + \frac{4 \cdot 3^x}{13^x} \geq 1 \]
This simplifies to:
\[ \left(\frac{5}{13}\right)^x + 4 \left(\frac{3}{13}\right)^x \geq 1 \]

### Step 3: Define a function
Let \( f(x) = \left(\frac{5}{13}\right)^x + 4 \left(\frac{3}{13}\right)^x \). We need to find when \( f(x) \geq 1 \).

### Step 4: Evaluate the function at specific points
1. **Evaluate at \( x = 1 \)**:
\[ f(1) = \left(\frac{5}{13}\right)^1 + 4 \left(\frac{3}{13}\right)^1 = \frac{5}{13} + \frac{12}{13} = \frac{17}{13} > 1 \]

2. **Evaluate at \( x = 2 \)**:
\[ f(2) = \left(\frac{5}{13}\right)^2 + 4 \left(\frac{3}{13}\right)^2 = \frac{25}{169} + 4 \cdot \frac{9}{169} = \frac{25 + 36}{169} = \frac{61}{169} < 1 \]

3. **Evaluate at \( x = 3 \)**:
\[ f(3) = \left(\frac{5}{13}\right)^3 + 4 \left(\frac{3}{13}\right)^3 = \frac{125}{2197} + 4 \cdot \frac{27}{2197} = \frac{125 + 108}{2197} = \frac{233}{2197} < 1 \]

### Step 5: Determine the solution set
From the evaluations:
- \( f(1) > 1 \)
- \( f(2) < 1 \)
- \( f(3) < 1 \)

This indicates that \( f(x) \) is decreasing. Therefore, the solution set for the inequality \( f(x) \geq 1 \) is:
\[ x \leq 1 \]

### Final Answer
The solution set for the inequality \( 5^x + (2\sqrt{3})^{2x} \geq 13^x \) is:
\[ (-\infty, 1] \]