Level Curve from a Lagrange multiplier problem: Depicted using Maple
Suppose we used the method of Lagrange multipliers to find maximal and minimal values for
f(x,y)=x^3-y^3 with respect to the constraint x^2+y^2=1. In doing so one obtains six points. Observe them in the picture, below.
In Maple type
with(plots, implicitplot); implicitplot([1 = x^2+y^2, -1 = x^3-y^3, -1/sqrt(2) = x^3-y^3, 1/sqrt(2) = x^3-y^3, 1 = x^3-y^3], x = -2..2, y = -2..2, color = [green, red, "Pink", blue, "Purple"], legend = [circle, k = -1, k = -sqrt(2)/2, k = sqrt(2)/2, k=1], gridrefine = 2)
to obtain:
Observe that there are 6 points of tangency, corresponding to the points found with the method of Lagrange multipliers.