Port563 <firstname.lastname@example.org> wrote: > This easy problem was very recently presented to me. > > Without any recourse to calculus, establish maximum and minimum values for: > > 7x + 4y > > where 2x^2 + y^2 + 2xy - 12x -8y -5 = 0, x and y being real. > > I found cheapo ways, e.g. > > * relying on inspection, then deducing the limits (tangents) of a set of > "-1.75 sloped" secants to an ellipse, etc.; > > alternatively > > * rejigging the expression's terms so it appears as a quadratic in y, > finding the discriminant and then simple manipulation to find/prove extrema; > > but, given the source of the problem, there might also be an elegant route I > missed. > > Anyone?
I do not know if it is elegant, but at least it does only need completing squares and no further calculus: try to rewrite the above equation 2x^2 + y^2 + 2xy - 12x -8y -5 = 0 as k(7x+4y) + S + r = 0 where k is a suitable nonzero integer, S is a sum of squares and r is a real number.