Find the greatest value of `x^3y^4` if `2x + 3y = 7` and `x>=0,y>=0` .

Explanation

To find the greatest value of \( x^3y^4 \) given the constraint \( 2x + 3y = 7 \) and \( x \geq 0, y \geq 0 \), we can use the method of Lagrange multipliers or apply the AM-GM inequality. Here, we will use the AM-GM inequality.

### Step-by-step Solution:

1. **Set Up the Problem**: We need to maximize \( z = x^3y^4 \) subject to the constraint \( 2x + 3y = 7 \).

2. **Express the Constraint**: From the constraint \( 2x + 3y = 7 \), we can express \( y \) in terms of \( x \):
\[
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