The solution set of `((3)/(5))^(x)=x-x^(2)-9` is

Options

A. {0}
B. {1}
C. `phi`
D. none of these

Answer

To solve the equation \(\left(\frac{3}{5}\right)^{x} = x - x^{2} - 9\), we will analyze both sides of the equation graphically and algebraically.

### Step 1: Define the functions
Let:
- \(y_1 = \left(\frac{3}{5}\right)^{x}\)
- \(y_2 = x - x^{2} - 9\)

### Step 2: Analyze \(y_1\)
The function \(y_1 = \left(\frac{3}{5}\right)^{x}\) is an exponential decay function because the base \(\frac{3}{5} < 1\).
- As \(x \to -\infty\), \(y_1 \to \infty\)
- As \(x \to \infty\), \(y_1 \to 0\)

### Step 3: Analyze \(y_2\)
The function \(y_2 = x - x^{2} - 9\) is a quadratic function. We can rewrite it as:
\[ y_2 = -x^{2} + x - 9 \]
This is a downward-opening parabola.

### Step 4: Find the vertex of \(y_2\)
To find the maximum value of the quadratic, we can use the vertex formula \(x = -\frac{b}{2a}\):
- Here, \(a = -1\) and \(b = 1\).
\[ x = -\frac{1}{2 \cdot -1} = \frac{1}{2} \]

### Step 5: Calculate the maximum value of \(y_2\)
Substituting \(x = \frac{1}{2}\) into \(y_2\):
\[ y_2\left(\frac{1}{2}\right) = \frac{1}{2} - \left(\frac{1}{2}\right)^{2} - 9 = \frac{1}{2} - \frac{1}{4} - 9 = \frac{1}{4} - 9 = \frac{1}{4} - \frac{36}{4} = -\frac{35}{4} \]

### Step 6: Analyze the intersection of the graphs
- The maximum value of \(y_2\) is \(-\frac{35}{4}\), which is negative.
- The function \(y_1\) approaches \(0\) as \(x\) increases, and is positive for all \(x\).

### Step 7: Conclusion
Since \(y_1\) is always positive and \(y_2\) has a maximum value that is negative, there are no points of intersection between the two graphs. Therefore, the solution set is empty.

### Final Answer
The solution set of the equation \(\left(\frac{3}{5}\right)^{x} = x - x^{2} - 9\) is:
\[ \text{Empty set} \quad \text{(denoted as } \emptyset \text{)} \]