What are the values of 'a' for which the function `f(x)=a^(x)` is :
(i) increasing
(ii) decreasing in R?

Answer

To determine the values of 'a' for which the function \( f(x) = a^x \) is increasing or decreasing in \( \mathbb{R} \), we need to analyze the derivative of the function.

### Step 1: Differentiate the function
The first step is to find the derivative of the function \( f(x) = a^x \).

\[ f'(x) = \frac{d}{dx}(a^x) = a^x \ln(a) \]

### Step 2: Determine when the function is increasing
A function is increasing when its derivative is greater than zero:

\[ f'(x) > 0 \implies a^x \ln(a) > 0 \]

Since \( a^x \) is always positive for any real \( x \) (as long as \( a > 0 \)), we only need to consider \( \ln(a) \):

\[ \ln(a) > 0 \]

This inequality holds when:

\[ a > 1 \]

### Step 3: Determine when the function is decreasing
A function is decreasing when its derivative is less than zero:

\[ f'(x) < 0 \implies a^x \ln(a) < 0 \]

Again, since \( a^x \) is always positive for \( a > 0 \), we need:

\[ \ln(a) < 0 \]

This inequality holds when:

\[ 0 < a < 1 \]

### Final Results
- The function \( f(x) = a^x \) is **increasing** for \( a > 1 \).
- The function \( f(x) = a^x \) is **decreasing** for \( 0 < a < 1 \).

### Summary of Values
- **Increasing:** \( a \in (1, \infty) \)
- **Decreasing:** \( a \in (0, 1) \)