Here is a matlab script

%%%%%%%%%%%%%%%%%%%%%%%
p1 = [0 0];
p2 = [2.6442 1.8984];
p3 = [.8567 .6468];
gradp1 = gradd(p1)
gradp2 = gradd(p2)
gradp3 = gradd(p3)
Hessp1 = eigsH(p1)
Hessp2 = eigsH(p2)
Hessp3 = eigsH(p3)

function Y  = gradd(X)

Y(1) = 20 * X(1)* X(2) - 10* X(1) - 4*X(1)^3;
Y(2) = 10 * X(1)^2 - 8*X(2) - 8*X(2)^3;
end

function Z = eigsH(X)
W = zeros(2,2);
W(1,1) = 20 * X(2) - 10 -12*X(1)^2;
W(1,2) = 20 * X(1);
W(2,1) = W(1,2);
W(2,2) = -8 - 24 * X(2)^2;
Z = eigs(W);
end



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
And the output from running it
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

gradp1 =

     0     0


gradp2 =

    0.0022   -0.0028


gradp3 =

   1.0e-03 *

    0.2231    0.2374


Hessp1 =

   -10
    -8


Hessp2 =

 -131.5028
  -18.9249


Hessp3 =

  -30.1381
    6.2265

%%%%%%%%%%%%%%%%%%%%%%%%%%

so p1 and p2 are local maxes and p3 is a saddle.