Question

Find a local minima and local maxima
f(x,y) = [cos(x+y)]^2 0<=x<=2*pi 0<=y<=2*pi

Answer 1

you can go \( \nabla f(x,y) = \nabla [cos^2(x+y)] = \\ <-\sin 2(x+y), -\sin 2(x+y)> = \vec 0 \implies 2(x+y) = 0, \pi, 2\pi, \dots\)


this BTW is the same as evaluating \(\dfrac{\partial f}{\partial x} = \dfrac{\partial f}{\partial y} = 0\) and i suspect you should do it that way instead but we should get the same outcome

so you'll get the lines of mins and max's, eg \(y = - x, y = \pi / 2 - x\) and you can see the pattern of peaks and troughs just by subbing in the actual values for x + y

can't remember OTOMH if these technically are actual max and min or actually saddles, because there is no actual maximum or minimum point, it's like lined up valleys and mountain tops.