Find the minimum value of `a x+b y` , where `x y=c^2` and `a ,\ b ,\ c` are positive.

Explanation

To find the minimum value of \( ax + by \) given the constraint \( xy = c^2 \), we can use the method of Lagrange multipliers or substitute the constraint into the function to minimize. Here, we will substitute the constraint.

### Step-by-Step Solution:

1. **Express \( y \) in terms of \( x \)**:
From the constraint \( xy = c^2 \), we can express \( y \) as:
\[
y = \frac{c^2}{x}
...