see this previous post
A plane with the equation (a,b,c > 0) together with the positive coordinate planes forms a tetrahedron of volume . Find the plane that minimizes V if the plane is constrained to pass through the point P = (1,1,1).
I'm not sure how to this, trying for and I got a=b=c. Which I don't think is right after using Lagrange multipliers. help!
= [(1/6)bc, (1/6)ac,(1/6)ab] and I set it to equal each other :/ help?
So your target function is and the constraint is .
so we must have , , and .
Since a specific value of is not a part of the solution, I find that, for many Lagrange multiplier problems, the best first step is to divide one equation by another, eliminating .
Here, dividing by we get so that and either a= b or a= -b. Since these numbers are positive, we must have a= b. Similarly, dividing the first equation by the third gives so that c= a.
That is, you were correct that a= b= c. And, of course, the condition that becomes and so a= b= c= 3.