Final answer:
To find the critical points of the function f(x,y) = xy(1-7x-9y), you can take the partial derivatives with respect to x and y and set them equal to zero. The critical points are (0,0), (1/7,0), (0,1/9), and (1/7,1/9).
Step-by-step explanation:
The critical points of the function f(x,y) = xy(1-7x-9y) can be found by taking the partial derivatives of f with respect to x and y and setting them equal to zero.
First, let's find the partial derivative of f with respect to x:
∂f/∂x = y - 14xy - 9y^2 = 0
Next, let's find the partial derivative of f with respect to y:
∂f/∂y = x - 7xy - 18xy = 0
Solving these equations simultaneously, we find the following critical points:
(0,0), (1/7,0), (0,1/9), and (1/7,1/9)