This is a problem from a Russian school math olympiad:
Minimize[(5 x^2 + 8 x*y + 5 y^2 - 14 x - 10 y +
30)/(4 - x^2 - 10 x*y - 25 y^2)^(7/2), {x, y}]
The above code is running on my comp as well as on MathematicaOnline without any response for hours. Also
NMinimize[(5 x^2 + 8 x*y + 5 y^2 - 14 x - 10 y +
30)/(4 - x^2 - 10 x*y - 25 y^2)^(7/2), {x, y}]
NMinimize::nrnum: The function value 0. +0.00126832 I is not a real number at {x,y} = {0.918621,0.716689}.
and
Plot3D[(5 x^2 + 8 x*y + 5 y^2 - 14 x - 10 y +
30)/(4 - x^2 - 10 x*y - 25 y^2)^(7/2), {x, -3, 3}, {y, -2, 2},PlotPoints->50 ]
do not help.
Solve[Thread[Numerator[Together[D[(5 x^2 + 8 x*y + 5 y^2 - 14 x - 10 y + 30)/(4 - x^2 - 10 x*y - 25 y^2)^(7/2), {{x, y}}]]] == 0], {x, y}, Reals]
. Then, check the solution withPositiveDefiniteMatrixQ[D[(5 x^2 + 8 x*y + 5 y^2 - 14 x - 10 y + 30)/(4 - x^2 - 10 x*y - 25 y^2)^(7/2), {{x, y}, 2}] /. First[%]]
. $\endgroup$