Chapter 13.9, Problem 4ES

(a) If you have a CAS with implicit plotting capability, use it to graph the circle ${\left(x-4\right)}^2+{\left(y-4\right)}^2=4$ and two level curves of $f\left(x,y\right)=x^3+y^3-3xy$ that just touch the circle.

(b) Use the result you obtained in part (a) to approximate the minimum value of f subject to the constraint ${\left(x-4\right)}^2+{\left(y-4\right)}^2=4.$

(c) Confirm graphically that you have found a minimum and not a maximum.

(d) Check your approximation using Lagrange multipliers and solving the required equations numerically.