Answer
a) saddle
b) local maximum
c) no information
Work Step by Step
Using the second derivative test:
a) compute D(a,b) = f_{xx}(1,1)f_{yy}(1,1)-[f_{xy}(1,1)]^2
D(0,2)= (-1)(1)-(6)^2=-37
since D(0,2)<0, g(0,2) is a saddle
b) compute D(a,b) = f_{xx}(1,1)f_{yy}(1,1)-[f_{xy}(1,1)]^2
D(0,2)= (-1)(-8)-(2)^2= 4 > 0
when D(a,b) > 0 , we have 2 cases dependent on f_{xx}(a,b)
so since f_{xx}(0,2) < -1 , g(0,2) is a local maximum
c) compute D(a,b) = f_{xx}(1,1)f_{yy}(1,1)-[f_{xy}(1,1)]^2
D(0,2)= (4)(9)-(6)^2=0
when D(a,b) = 0 then the test gives no information
