find all local max,min and/or saddle point of f(x,y)=4+x^3+y^-3xy

Question

anonymous · Answer

would you plz write the function using equation

amistre64 · Answer

hmm, something to do with:$$f_{xy}f_{yx}-(f_{xx})^2$$
or did i recall that incorrectly?

amistre64 · Answer

almost had it :)
$$f_{xx}f_{yy}-(f_{xy})^2$$

anonymous · Answer

$$4+x ^{3}+y ^{3}-3xy$$

anonymous · Answer

it fxx.fyy-(fxy)^2

anonymous · Answer

i jst want to verify the answer for it !

turingtest · Answer

what answer did you get?

anonymous · Answer

$$fx = 2x-3y $$
$$fy= 2y-3x$$
on solving both equations the point is (0,0)
$$fxx = 2$$ , $$fyy= 2$$ $$fxy = $$
$$D = -5 <0 saddle point$$

turingtest · Answer

that's not quite right I don't think

anonymous · Answer

fxy = -3

turingtest · Answer

$$f(x,y)=4+x^3+y^3-3xy$$$$f_x=3x^2-3y=0\implies y=x^2$$$$f_y=3y^2-3x=3x^4-3x=3x^3(x-1)=0$$$$ x=\{0,1\},y=\{0,1\}$$

anonymous · Answer

its x^3
so i have (0,0) ,(1,1)and a (-1,1) 
i am not sure if (-1,1) should be there oor not

anonymous · Answer

@Living_dreams  I solved what you mentioned in your comment

turingtest · Answer

there is no x=-1 solution, I don't see how you get that

turingtest · Answer

@Avva I don't see how you get your fx and fy

anonymous · Answer

Ooh I saw the powers on X ^ y 2 not 3

anonymous · Answer

@Avva thanks for ur help but i was wonder abt the max and min points!  n\ @TuringTest : 3y^2-3x=0 i sub y=x^2 from other equation so it will be 3x(x^3-1)=0
therefore you have x =0,1,-1
i am not sure abt -1 though !! tht was my question

turingtest · Answer

x^3=1 does not have x=-1 as a solution since (-1)^3=-1, not 1

turingtest · Answer

I did screw up my factoring earlier though, you were right about that :P

turingtest · Answer

3x(x^3-1)=0
3x=0 -> x=0
x^3=1 -> x=1

anonymous · Answer

@TuringTest that alrite!!  :)
 so now my question is x^3=1 so will i hv x=-1 if i solve for x ?

turingtest · Answer

no, again, because $(-1)^3\neq1$ 
so $x^3=1$ has only one real solution, x=1
x=-1 is NOT a solution

anonymous · Answer

the other two roots are complex not real

anonymous · Answer

@TuringTest  oh ok thanks a lot for ur help!! 
so the final ans will be (0,0 ) is a saddle point and (1,1) will be a local minimum !

turingtest · Answer

exactly :)

anonymous · Answer

@Avva :thanks for ur help !! :)

anonymous · Answer

@Living_dreams  URW :) sorry I didn't get that the powers are 3 at first

anonymous · Answer

it okay !!i knw it get hard to tell the difference between 2 and 3 if we type it in equation form !!