I have a math assignment due this week and I really need help. I dont really understand the professor in class because he uses big words and explains in a very complex manner.
The question is:
The cylinder $x^2 + y^2 = 8$ intersects the plane $x + y + z = 1$ to form an ellipse.
i) Construct the Lagrangian, L, for finding the maximum and minimum values of $f(x, y, z) = 4 - z$ on the ellipse.
ii) Find the stationary points for L.
iii) Hence, find the maximum and minimum values of $f(x, y, z)$.
I hope someone can help me because I am really so lost.
We have $L(x,y,z, \lambda, \mu)=4-z+\lambda(x^2+y^2-8)+ \mu(x+y+z-1)$.
For the stationary points of $L$ you have to solve the system
$2 \lambda x+\mu=0$,
$2 \lambda y+\mu=0$,
$-1+\mu=0$,
$x^2+y^2=8$,
$x+y+z=1$.
Your turn !