If `ab=2a+3b, agt0, b gt0`, then the minimum value of ab is

Options

A. 12
B. 24
C. `(1)/(4)`
D. none of these

Explanation

To find the minimum value of \( ab \) given the equation \( ab = 2a + 3b \) with the constraints \( a > 0 \) and \( b > 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the equation:
\[ ab = 2a + 3b \]
Rearranging gives:
\[ ab - 2a - 3b = 0 \]
This can be rewritten as:
\[ b(a - 3) = 2a \]
From this, we can express \( b \) in terms of \( a \):
\[ b = \frac{2a}{a - 3} \]

### Step 2: Substitute \( b \) into \( ab \)
Now substitute \( b \) back into the expression for \( ab \):
\[ ab = a \cdot \frac{2a}{a - 3} = \frac{2a^2}{a - 3} \]
Let \( z = ab \), so:
\[ z = \frac{2a^2}{a - 3} \]

### Step 3: Differentiate to Find Critical Points
To find the minimum value, we need to differentiate \( z \) with respect to \( a \):
\[ \frac{dz}{da} = \frac{(2a^2)'(a - 3) - (2a^2)(a - 3)'}{(a - 3)^2} \]
Calculating the derivatives:
\[ (2a^2)' = 4a \quad \text{and} \quad (a - 3)' = 1 \]
Thus, we have:
\[ \frac{dz}{da} = \frac{4a(a - 3) - 2a^2}{(a - 3)^2} \]
Simplifying the numerator:
\[ = \frac{4a^2 - 12a - 2a^2}{(a - 3)^2} = \frac{2a^2 - 12a}{(a - 3)^2} \]

### Step 4: Set the Derivative to Zero
Setting the derivative equal to zero to find critical points:
\[ 2a^2 - 12a = 0 \]
Factoring out \( 2a \):
\[ 2a(a - 6) = 0 \]
This gives us:
\[ a = 0 \quad \text{or} \quad a = 6 \]
Since \( a > 0 \), we take \( a = 6 \).

### Step 5: Find Corresponding \( b \)
Now substitute \( a = 6 \) back into the equation for \( b \):
\[ b = \frac{2(6)}{6 - 3} = \frac{12}{3} = 4 \]

### Step 6: Calculate Minimum Value of \( ab \)
Now calculate \( ab \):
\[ ab = 6 \cdot 4 = 24 \]

### Conclusion
Thus, the minimum value of \( ab \) is:
\[ \boxed{24} \]