Chapter 8.4, Problem 20E

Consider the following constrained optimization problem:

Minimize $f\left(x,y,z\right)=x^2+{\left(y+3\right)}^2+{\left(z-4\right)}^2$

Subject to $4-x^2-y^2-z=0$.

a. Explain why this minimization problem must have a solution, and solve it using the method of Lagrange multipliers.

b. Solve it again using the substitution method by solving the constraint equation for z.

c. Now try to solve it using the substitution method by solving the constraint equation for x.

d. Explain what goes wrong in part (c).