If ` ax+(b)/(x) ge c, AA a gt 0 and a,b,c ` are positive constant then

Options

A. `ab ge (c^(2))/(4)`
B. `ab le (c^(2))/(4)`
C. `bc ge (a^(2))/(4)`
D. `ac ge (b^(2))/(4)`

Answer

To solve the inequality \( ax + \frac{b}{x} \geq c \) where \( a, b, c \) are positive constants and \( x > 0 \), we can follow these steps:

### Step 1: Rearrange the Inequality
Start by rewriting the inequality:
\[ ax + \frac{b}{x} - c \geq 0 \]
This can be expressed as:
\[ ax + \frac{b}{x} \geq c \]

### Step 2: Multiply through by \( x \)
Since \( x > 0 \), we can multiply both sides by \( x \) without changing the direction of the inequality:
\[ ax^2 + b - cx \geq 0 \]
Rearranging gives us:
\[ ax^2 - cx + b \geq 0 \]

### Step 3: Analyze the Quadratic
The expression \( ax^2 - cx + b \) is a quadratic in \( x \). For this quadratic to be non-negative for all \( x > 0 \), we need to check two conditions:
1. The coefficient of \( x^2 \) (which is \( a \)) must be positive.
2. The discriminant of the quadratic must be less than or equal to zero.

### Step 4: Calculate the Discriminant
The discriminant \( D \) of the quadratic \( ax^2 - cx + b \) is given by:
\[ D = (-c)^2 - 4ab = c^2 - 4ab \]
For the quadratic to be non-negative for all \( x \), we require:
\[ D \leq 0 \implies c^2 - 4ab \leq 0 \]
This simplifies to:
\[ 4ab \geq c^2 \]

### Step 5: Final Condition
From the above inequality, we can derive:
\[ ab \geq \frac{c^2}{4} \]
Thus, we conclude that the condition for the inequality \( ax + \frac{b}{x} \geq c \) to hold for all \( x > 0 \) is:
\[ ab \geq \frac{c^2}{4} \]

### Conclusion
Since \( a, b, c \) are positive constants, the correct option based on the derived condition is:
\[ ab \geq \frac{c}{4} \]