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
(iii) 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 will evaluate the statements given in the question.

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

1. **Understanding the behavior of the function**:
- The base \( a \) is between 0 and 1. This means that as \( x \) increases, \( a^x \) will decrease because raising a fraction (less than 1) to a higher power results in a smaller number.

### Step 2: Determine if the function is increasing or decreasing

2. **Check if the function is increasing or decreasing**:
- To determine if \( f(x) \) is increasing or decreasing, we can look at the derivative \( f'(x) \).
- The derivative of \( f(x) = a^x \) is given by:
\[ f'(x) = a^x \ln(a) \]
- Since \( 0 < a < 1 \), \( \ln(a) < 0 \). Therefore, \( f'(x) < 0 \) for all \( x \).
- This indicates that \( f(x) \) is a decreasing function.

### Step 3: Check the point of intersection at (0, 1)

3. **Evaluate the function at \( x = 0 \)**:
- We can check the value of the function at \( x = 0 \):
\[ f(0) = a^0 = 1 \]
- This shows that the function passes through the point \( (0, 1) \).

### Conclusion

From the analysis, we can conclude:
- The function \( f(x) = a^x \) is a decreasing function.
- The function intersects at the point \( (0, 1) \).

Thus, the correct statements are:
- (ii) It is a decreasing function.
- (iii) All such functions must meet at (0, 1).

The correct option is **D: 2 and 3 are true**.