Thread: Calculus 3 - Optimization without Constraints

The behavior of a function can be complicated near a critical point where D=0. Suppose that f(x,y)= x^3 - 3xy^2.
a.) Show that there is one critical point at (0,0) and that D=0 at that point.
b.) Show that the contour for f(x,y)=0 consists of three lines intersecting at the origin where f alternates from positive to negative. Sketch a contour diagram for f near 0.

I already did part a, but I have no idea how to even start part b.

Originally Posted by alyssalynnx38

The behavior of a function can be complicated near a critical point where D=0. Suppose that f(x,y)= x^3 - 3xy^2.
a.) Show that there is one critical point at (0,0) and that D=0 at that point.

firstly solve for

Like I said, I already did part a.







If you set fx and fy equal to 0, both x and y are 0 because xy = 0 and x^2 + y^2 = 0





So, there is a critical point at (0,0) and D = 0. And that's as far as I was able to get.