3x + 4y $$ \le $$ 100

4x + 3y $$ \le $$ 75

x $$ \ge $$ 0, y $$ \ge $$ 0

Feasible region is shown in the graph

Let maximum value of 6xy + y² = c

For a solution with feasible region,

6xy + y² = c and 4x + 3y = 75 must have at least one positive solution.

$${y^2} + 6y\left( {{{75 - 3y} \over 4}} \right) - c = 0 $$

$$\Rightarrow {7 \over 2}{y^2} - {{225} \over 2}y + c = 0$$

$$ \Rightarrow {\left( {{{225} \over 2}} \right)^2} \ge 4.{7 \over 2}.c $$

$$\Rightarrow c \le {{{{225}^2}} \over {56}} \approx 904$$