Find all the points where these functions have any local extrema?

Question

find all the points where these functions have any local extrema, give the values of any local extrema, identify any saddle points

function is
f(x,y)=2y^3+5x^2+60xy+25

the ending is whats giving me a hard time
what do i do after
y = 0, 60
x = 0, -360
or is that it?

Answer 1

fx=10x+60y=0
fy=6y^2+60x=0=>y^2+10x=0 so 10x=-y^2 and
-y^2+60y=0 which gives y=0 and y=60
for y=0 x=0
for y=60 x=-360
critical points (0,0) and (-360,60)
Now we must study the Hessian fxx*fyy-fxy^2 = H
fxx=10
fyy=12y
fxy=60
At(0,0) H=-3600<0 so this is a saddle point
at(-360,60) H = 10(720)-3600= 3600>0 and fxx=10>0
so this is a local minimum