Question

(1 point) Consider the function f(x, y) = 4x+ + 8y. List all critical points of f(x, y). If there are none, enter "none". If there is more than one, enter a comma-separated list of ordered pairs, e.g., "(1,2), (3,4). (5,6)" Critical points are List all critical points of f(x,y) which are local maxima. If there are none, enter "none". If there is more than one, enter a comma-separated list of ordered pairs, e... "(1,2), (3,4), (5,6) Local maxima occur at List all critical points of f(x,y) which are local minimo. If there are none, enter "none". If there is more than one, enter a comma-separated list of ordered pairs, e... "(1,2), (3,4), (5,6)" Local minima occur at List all critical points of (x,y) which are saddle points. If there are none, enter "none". If there is more than one, enter a comma-separated list of ordered pairs, 6.9. "(1,2), (3,4), (5,6)". Saddle points occur at

Answer 1

Solution: f(x,y) = 4x4 + 8y4 of дх = 16x3 = 0 ............... (1) af 32y3 = 0 ....... .(2) ду Solving the equations (1) and (2) Critical points are (x,y) = (0,0) r= 22f дх2 = 48x2 22f t= 96y2 dy2 S= af дхду = 0 D = rt - 52 48 x 96 xay2 At the point (0,0) rt -s2 = 0, r = 0 Hence it is a doubtful case. It needs further investigations. There is a global minimum at (0,0) Hence a local maximum occurs at None Hence a local minimum occurs at None Saddle point occurs at None