Find and classify all the critical points for f(x,y)=3x^2y-y^3-3x^2+2

cistG 2021-03-04 Answered
Find and classify all the critical points for f(x,y)=3x2yy33x2+2
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Laith Petty
Answered 2021-03-05 Author has 103 answers

Let z=f(x). Differentiating wrt x and y:
dzdx=6xy+3x2dydx3y2dydx6x,
dzdy=6xydxdy+3x23y26xdxdy.
dzdx=3(x2y2)dydx+6x(y1),
dzdy=6x(y1)dxdy+3(x2y2).
The general form is: dzdt=6xydxdt+3x2dydt3y2dydt6xdxdt, where t is a parameter such that x=g(t), y=h(t) and z=j(t), where g, h and j are functions of t.
When dzdx=0 or dzdy=0 there is a critical point: dydx=2xy1y2x2.
The general form is dzdt=6x(y1)dxdt+3(x2y2)dydt=0.
So x=y=1 is a critical point, x=y=0 is another, x=-1 and y=1 is another.
When x=y=1+d, where d is very small, dzdt=6d(1+d)dxdt+0 which is positive when d and dx/dt are both positive or both negative. This suggests a minimum at x=y=1. If x=y=d,

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more