
Critical point question
hi everyone
having problems finding critical points for this two equation to fined the maximum,minimum or saddle points.
a) $\displaystyle x^4+y^44xy+1$
$\displaystyle f_x=4x^34y$, $\displaystyle f_y=4y^34x$
b) f(x,y)=$\displaystyle x^42x^2y+2y$
$\displaystyle f_x=4x^34xy$,$\displaystyle f_y=22x^2$
need help to find the critical points for this two equations,really hope someone can help me. really appreciate all your help & support.

How far have you come in your attempts? Where do critical points for a given function occur?

thank you for replying.
a) $\displaystyle x^3y=0$,$\displaystyle y^3x=0$
$\displaystyle x^3=y$
$\displaystyle y^3=x$
im stuck here, unable to find the critical point
b)$\displaystyle 4x^34xy=0$
$\displaystyle 4x^3=4xy$
$\displaystyle x^2=y$
$\displaystyle 22x^2=0$
$\displaystyle 2=2x^2$
x=1 & 1
critical point, = (1,1) & (1,1) ..is this right???
realyl hope someone can help, appreciate all help & support.

a: Remember that the two derivatives form a system of equations:
$\displaystyle x^3y = 0$
$\displaystyle y^3x = 0$
imply that
$\displaystyle x^9=x$ or $\displaystyle x(x^81) = 0$ (see why?)
What values for critical points does this give you?
b: Seems right to me, so far, however, you are being a little hasty. It's quite right that
$\displaystyle 4x^34xy = 0$ implies that
$\displaystyle x^2 = y$
but it also implies that
$\displaystyle x(x^2y) = 0$. See what you missed?

thank you for replying,
a)
$\displaystyle (x^3)^3x=0$
$\displaystyle x(x^81)$
x= 1,0 & 1
critical points=(1,1)(0,0) & (1,1)
b)you are right, but is my final answer for critical point correct?
critical point=(1,1) & (1,1) .. i think it has to be correct..not sure,a bit confused though
is this correct?
thank you again for all help & support.