A function is defined as `f(x)=a^x AA x in R` and `0ltalt1`, then which of the following is true?
(i) It is an increasing function
(ii) It is a decreasing function
All such functions must meet at (0,1)

Options

A. only (i) is true
B. only (ii) is true
C. (i) and (iii) are true
D. (ii) and (iii) are true

Answer

To solve the problem, we need to analyze the function \( f(x) = a^x \) where \( a \) is a constant such that \( 0 < a < 1 \). We want to determine whether this function is increasing or decreasing.

### Step-by-Step Solution:

1. **Define the Function**:
We have the function defined as:
\[ f(x) = a^x \]
where \( a \) is a constant in the interval \( (0, 1) \).

2. **Differentiate the Function**:
To determine if the function is increasing or decreasing, we need to find its derivative:
\[ f'(x) = \frac{d}{dx}(a^x) = a^x \ln(a) \]

3. **Analyze the Derivative**:
- Since \( 0 < a < 1 \), the natural logarithm of \( a \) will be negative:
\[ \ln(a) < 0 \]
- The term \( a^x \) is always positive for any real number \( x \):
\[ a^x > 0 \]

4. **Determine the Sign of the Derivative**:
- Now, we can analyze the sign of the derivative:
\[ f'(x) = a^x \ln(a) \]
- Since \( a^x > 0 \) and \( \ln(a) < 0 \), it follows that:
\[ f'(x) < 0 \]
- This implies that the derivative is negative for all \( x \).

5. **Conclusion**:
- Since the derivative \( f'(x) < 0 \) for all \( x \), we conclude that the function \( f(x) = a^x \) is a decreasing function.

6. **Point of Intersection**:
- All such functions \( f(x) = a^x \) for different values of \( a \) (where \( 0 < a < 1 \)) will intersect at the point \( (0, 1) \) because:
\[ f(0) = a^0 = 1 \]
- Therefore, all these functions meet at the point \( (0, 1) \).

### Final Answer:
- The correct statements are:
- (ii) It is a decreasing function.
- All such functions must meet at (0, 1).