Which point of the mountain A mountaineer wishes to make a descent from a mountain, starting at the top of the mountain at $(0, 0)$. The height of the mountain at position (x, y) is given by $$h(x, y) = 3000 - \frac{1}{10000}(5x^2+4xy+2y^2)$$ In 30 minutes, the mountaineer can arrive at any point that lies on a circle with a radius 1000 and centered at $(0, 0)$. Which point should the mountainer target if he is to descent as much as possible? Using Lagrange function

You can simplify the algebra a bit by realizing that you are really just looking to maximize $$f(x,y) = 5x^2+2xy+2y^2 $$ subject to the constraint $$g(x,y)=x^2+y^2=1000$$ so that $$f(x,y) = 3x^2+2xy+2000 $$