Suppose we seek to optimize the objective function $f(x, y)$ subject to a constraint of the form $g(x, y)=0$ The resulting Lagrange Function is given by $F(x, y, \lambda)=9 x-\frac{64}{y}+\lambda g(x, y)$. We proceed to find four critical points: $(2,8)$ $(-2,8)$ $(2,-8)$ $(-2,-8)$ What is the value ("height") of the objective function at its relative maximum? Max value of $f(x, y)$ is

Step 1 ：Suppose we seek to optimize the objective function \(f(x, y)\) subject to a constraint of the form \(g(x, y)=0\)

Step 2 ：The resulting Lagrange Function is given by \(F(x, y, \lambda)=9 x-\frac{64}{y}+\lambda g(x, y)\)

Step 3 ：We proceed to find four critical points: \((2,8)\), \((-2,8)\), \((2,-8)\), and \((-2,-8)\)

Step 4 ：To find the maximum value of the function \(f(x, y)\), we need to substitute the critical points into the function and find the maximum value among them

Step 5 ：The critical points are given as \((2,8)\), \((-2,8)\), \((2,-8)\), and \((-2,-8)\)

Step 6 ：We need to substitute these points into the function \(f(x, y) = 9x - \frac{64}{y}\) and find the maximum value

Step 7 ：The values of the function at the critical points are \(10.0\), \(-26.0\), \(26.0\), and \(-10.0\)

Step 8 ：The maximum value of the function \(f(x, y)\) at its relative maximum is \(\boxed{26}\)