Chapter 14.8, Problem 2E

To determine

We need to find the extreme values of $f\left(x,y\right)=x^2+2y^2$ subject to the constraint $g\left(x,y\right)=4x-6y=25$.

a. Show that the Lagrange equations yield $2x=4\lambda ,4y=-6\lambda$.

b. Show that if $x=0\text{ or }y=0$, then the Lagrange equations give $x=0\text{ y=0}$.Since $\left(0,0\right)$ does not satisfy the constraint, you may assume that x and y are nonzero.

c. Use the Lagrange equations to show that $y=-\frac{3}{4}x$.

d. Substitute in the constraint equation to show that there is a unique critical point P.

e. Does P correspond to a minimum or maximum value of f?