If `5- (3a-2b)^(2)` will be `"max"^(m)`, when `(a)/(b)`= ?

Answer

To solve the problem, we need to maximize the expression \( 5 - (3a - 2b)^2 \). The key to maximizing this expression is to minimize the term \( (3a - 2b)^2 \).

### Step-by-step Solution:

1. **Understanding the Expression**:
The expression we want to maximize is \( 5 - (3a - 2b)^2 \). To maximize this expression, we need to minimize the term \( (3a - 2b)^2 \).

2. **Minimizing the Square Term**:
The term \( (3a - 2b)^2 \) is a square of a real number. The minimum value of a square term is \( 0 \) (since squares cannot be negative). Therefore, we set:
\[ 3a - 2b = 0 \]

3. **Solving for the Ratio \( \frac{a}{b} \)**:
From the equation \( 3a - 2b = 0 \), we can rearrange it to find the relationship between \( a \) and \( b \):
\[ 3a = 2b \]
Dividing both sides by \( b \) (assuming \( b \neq 0 \)):
\[ \frac{a}{b} = \frac{2}{3} \]

4. **Conclusion**:
The value of \( \frac{a}{b} \) that maximizes the expression \( 5 - (3a - 2b)^2 \) is:
\[ \frac{a}{b} = \frac{2}{3} \]