Let a, b and c be fixed positive real numbers. Let `f (x)=(ax)/(b+cx)` for `x le 1`. Then as x increases,

Options

A. f (x) increases
B. f (x) decreases
C. f (x) increases first and then decreases
D. None of these

Answer

To analyze the function \( f(x) = \frac{ax}{b + cx} \) for \( x \leq 1 \) and determine how it behaves as \( x \) increases, we can follow these steps:

### Step 1: Identify the function and its domain
The function is given as:
\[ f(x) = \frac{ax}{b + cx} \]
with the condition that \( x \leq 1 \). Here, \( a, b, c \) are fixed positive real numbers.

**Hint:** Understand the structure of the function and the constraints on \( x \).

### Step 2: Determine the behavior of \( f(x) \) as \( x \) approaches 1
First, we can evaluate the function at the endpoint of the domain, which is \( x = 1 \):
\[ f(1) = \frac{a(1)}{b + c} = \frac{a}{b + c} \]
Since \( a, b, c > 0 \), we know that \( f(1) > 0 \).

**Hint:** Calculate the function value at the boundary of the domain to understand the endpoint behavior.

### Step 3: Analyze the function as \( x \) approaches 0
Next, we evaluate the function at \( x = 0 \):
\[ f(0) = \frac{a(0)}{b + c(0)} = 0 \]
This indicates that the function starts at 0 when \( x = 0 \).

**Hint:** Check the function value at another point in the domain to see how it behaves at the start.

### Step 4: Investigate the derivative to determine monotonicity
To analyze whether the function is increasing or decreasing, we can find the derivative \( f'(x) \):
Using the quotient rule:
\[ f'(x) = \frac{(b + cx)(a) - (ax)(c)}{(b + cx)^2} \]
Simplifying the numerator:
\[ f'(x) = \frac{ab + acx - acx}{(b + cx)^2} = \frac{ab}{(b + cx)^2} \]
Since \( a, b > 0 \) and \( (b + cx)^2 > 0 \) for \( x \leq 1 \), we conclude that \( f'(x) > 0 \) for all \( x < 1 \).

**Hint:** Use the derivative to determine whether the function is increasing or decreasing.

### Step 5: Conclusion
Since \( f'(x) > 0 \) for all \( x < 1 \), the function \( f(x) \) is increasing as \( x \) increases from 0 to 1. Therefore, the correct answer to the question is that \( f(x) \) increases as \( x \) increases.

**Final Answer:** \( f(x) \) increases as \( x \) increases for \( x \leq 1 \).