If `f(x)=((3)/(5))^(x)+((4)/(5))^(x)-1,x inR,` then the equation `f(x)=0` has :

Options

A. no solution
B. one solution
C. two solutions
D. more than two solutions

Answer

To solve the equation \( f(x) = 0 \) where \( f(x) = \left( \frac{3}{5} \right)^x + \left( \frac{4}{5} \right)^x - 1 \), we will analyze the function step by step.

### Step 1: Analyze the function \( f(x) \)

The function is defined as:
\[ f(x) = \left( \frac{3}{5} \right)^x + \left( \frac{4}{5} \right)^x - 1 \]

### Step 2: Determine the behavior of \( f(x) \) as \( x \to \infty \)

As \( x \) approaches infinity:
\[ \left( \frac{3}{5} \right)^x \to 0 \quad \text{and} \quad \left( \frac{4}{5} \right)^x \to 0 \]
Thus,
\[ f(x) \to 0 + 0 - 1 = -1 \]

### Step 3: Determine the behavior of \( f(x) \) at \( x = 0 \)

At \( x = 0 \):
\[ f(0) = \left( \frac{3}{5} \right)^0 + \left( \frac{4}{5} \right)^0 - 1 = 1 + 1 - 1 = 1 \]

### Step 4: Differentiate \( f(x) \)

To analyze the monotonicity of \( f(x) \), we differentiate it:
\[ f'(x) = \frac{d}{dx} \left( \left( \frac{3}{5} \right)^x \right) + \frac{d}{dx} \left( \left( \frac{4}{5} \right)^x \right) \]
Using the derivative of \( a^x \) which is \( a^x \ln(a) \):
\[ f'(x) = \left( \frac{3}{5} \right)^x \ln\left( \frac{3}{5} \right) + \left( \frac{4}{5} \right)^x \ln\left( \frac{4}{5} \right) \]

### Step 5: Analyze the sign of \( f'(x) \)

Since \( \frac{3}{5} < 1 \) and \( \frac{4}{5} < 1 \), both \( \ln\left( \frac{3}{5} \right) \) and \( \ln\left( \frac{4}{5} \right) \) are negative. Therefore, \( f'(x) < 0 \) for all \( x \).

### Step 6: Conclusion on the monotonicity

Since \( f'(x) < 0 \), the function \( f(x) \) is monotonically decreasing.

### Step 7: Identify the roots

We have determined:
- \( f(0) = 1 > 0 \)
- \( f(x) \to -1 < 0 \) as \( x \to \infty \)

Since \( f(x) \) is continuous and decreases from a positive value at \( x = 0 \) to a negative value as \( x \to \infty \), by the Intermediate Value Theorem, there must be exactly one root in the interval \( (0, \infty) \).

### Final Answer

The equation \( f(x) = 0 \) has exactly **one root**.

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