Question

find all the critical points of the following function. indicate whether each point is a local max, min or a saddle point.\[f(x,y)=x ^{3}-y ^{3}-2xy+6\]

Answer 1

\[\max =\frac{170}{27}\text{ at }(x,y)=\left(-\frac{2}{3},\frac{2}{3}\right)\]

Answer 2

would you mind to show your working on obtaining \[(\frac{-2}{3}, \frac{2}{3})\]? because when i tried to find the value of y when x is -2/3, i got +-2/3 for y; and when i use (-2/3, 2/3) to determine max or mine, the discriminant D equals to 0.

Answer 3

\[D=f _{xx}f _{yy}-f _{xy }^{2}\]