I am trying to solve the following exercise : Use lagrange multiplier to find the critical points of Z $Z=x(y+4)$ constraint $x+y=8$ Then find if a relaxation of the constraint will increase or reduce the value of Z.
I found that the critical point is (x,y)=(6,2) with λ=6 But I don't know what is the relaxation ?
Could you help me please ? Is it based on the value of λ if it is positive or negative ?
Thanks in advance
The Lagrangian is given by $$L = x(y+4) + \lambda [8-x-y] $$
F.O.C. are given by $$ y+4-\lambda=0 \\ x-\lambda=0 $$
Finally, we know that $$ x+y=8 $$
Solving this system we find that $x=6$, $y=2$ and $\lambda=6$ as you correctly found. The value of Z at the point $(x,y)=(6,2)$ is 36.
That Lagrange multiplier is positive means that relaxing constraints will lead to an increase in the value of Z. In other words, imagine that the constraint was $x+y=a$ for some $a>0$. Then if you find that at a given value of $a$, in your case $a=8$, $\lambda>0$ then it means that increasing $a$ would lead to a new solution at which the value of Z will be higher than before.
Note: It is a little bit strange to talk about relaxing the constraints in the case of equality constraints. Typically, such language makes sense if the constraint is written as inequality, for example in your case as $x+y \leq 8$.