First of this is somewhat embarassing as I really shouldn't be strugling with this, however I'm having a hard time getting back into this after a break of about a year.
I'm trying to obtain the critical points of the function bellow and then determin their nature. I'm having no trouble with the differentiation however solving the first derivatives for x and y is driving me nuts. Once I have these values I know how to solve the rest of the problem.
A surface has the following equation:
dz/dx= 3x^2-3y = 0
dz/dy= -3x+24y^2 = 0
3x^2=3y so x^2=y
24y^2=3x so 8y^2=x
x = (A)??? & (B)???
y = (A)??? & (B)???
Any help is greatly appreciated.
You have a critical point when the tangent plane is "vertical", so first you find the equation for your tangent plane:
then the tangent plane is
, and the plane is vertical when both are equal to zero.
So all you need to do is decide when
Using the second equation and squaring we get:
and substituting into the first you get:
, so we get
which works in the original set of equations, otherwise
, this gives the other option as:
from the first equation, and checking with the second equation, we see that only the positive possibility works.