I have problems with question in supplemental problem set 2. In, F-3 the product of the intercepts are D = c^3/(ab) where c = 1-2a-b and where z = ax + by + c. The partial derivative of D with respect to a I found to be b(b-4a-1)(1-2a-b)^2 =0 and the derivative of D with respect to b I found to be a(2a-2b-1)(1-2a-b)^2 = 0. How do I proceed to find out the values of a, b and c that give the minimum product of the intercepts?

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I have problems with question in supplemental problem set 2. In, F-3 the product of the intercepts are D = c^3/(ab) where c = 1-2a-b and where z = ax + by + c. The partial derivative of D with respect to a I found to be b(b-4a-1)(1-2a-b)^2 =0 and the derivative of D with respect to b I found to be a(2a-2b-1)(1-2a-b)^2 = 0. How do I proceed to find out the values of a, b and c that give the minimum product of the intercepts?

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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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would like to help but have no access to original question
  • phi
In the form z= ax + by+ c the (stipulated positive) intercepts on the 3 axes occur at c, -c/a, -c/b thus c>0, a<0, b<0 we have two (ugly) equations and 2 unknowns. \[ b(b-4a-1)(1-2a-b)^2 =0 \\ a(2a-2b-1)(1-2a-b)^2 = 0\] a,b,c are not zero. As the (1-2a-b) term is in both equations, having it be 0 adds no information, thus I would solve using \[ (b-4a-1)=0 \\(2a-2b-1) = 0\] and along with \[ c = 1-2a-b \] we can solve for a, b, and c in z= ax + by+ c that minimize the the product of the intercepts.
I did that as well, and my answer is a = -1/2 b = -1 and c = 3. Using Lagrange multipliers, I arrive at the same answer. Thanks!

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