Chapter 14.8, Problem 1E

To determine

We need to find the extreme values of the function $f\left(x,y\right)=2x+4y$ subject to the constraint $g\left(x,y\right)=x^2+y^2-5=0$.

a. Show that the Lagrange equation $\nabla f=\lambda \nabla g$ gives $\lambda x=1$ and $\lambda y=2$

b. Show that these equations imply $\lambda \ne 0$ and $y=2x$.

c. Use the constraint equation to determine the possible critical points $\left(x,y\right)$.

d. Evaluate $f\left(x,y\right)$ at the critical points and determine the minimum and maximum values.