1. find all critical pts 2.classify each as location of rel max, min or saddle pt F(x,y)=x3-y2-3x+6y?

Answer 1

gradient of F = (3x^2-3, -2y+6 ) =(0,0),
if 3x^2-3=0 and -2y+6=0
so
x^2=1 and y=3
(x=1 or x=-1) and y=3
so there are 2 points:
(1,3) and (-1,3)

f_xx = 6x
f_xx f_yy -(f_xy)^2 = 6x(-2) -(0 )^2=-12x

point (1,3):
f_xx (1,3)f_yy(1,3) -(f_xy(1,3))^2 = -12(1)<0 so (1,3) is a saddle point.

point (-1,3):
f_xx(-1,3)=6(-1)<0
and
f_xx (-1,3)f_yy(-1,3) -(f_xy(-1,3))^2 = -12(-1) >0, so (-1,3) is a local maximum

Answer 2

I onl;y tell yu the steps.

1) the critical points are solutions of dF(x,y) = 0

dF(x,y) is a matrix 2x2

..... look in your book this textbox is not graphic enough to show you the solution...