Question

[3 ] Consider the cost function given by * = r cos(o), y = r sin(e), r = V22 + y2, tan(0) = G(ro), or, G(x, y) =r(1-r)sin(70) - Use the first gradient condition to obtain the extremal points, and use the second deriva- tive condition to determine wether the point is a max, min, or a saddle point.

Would like help solving this problem. This can be left in terms of r and theta or converted to x and y. Using MATLAB (the computer program) may be helpful.

Answer 1

x= r Con O ga s sino Gle, y) = r (1-0) Sin ito G (ro) = o(1-r) Sinto Go Cr,0) = r Sin Toort Simko G (ro) = Sim To-28 Sinto G" (ro) = 1 - 2 sinto G(role o - 2 sinto = 0 Sinto=0 Wheren EZ

All integers are the extremal points, now to determine maxima/minima just substitute the values of theta in G(r,theta) and see if the expression is positive or negative or zero. If it's positive the point is minima , if negative the point is maxima and if zero it's a saddle point.