mathematical optimization - Why does Maximize return a wrong answer?

I want to find the maximum of the following function, but get a wrong result. What am I doing wrong?

Clear[x, y, xx, yy]
peaks[x_, y_] := 
  3*(1 - x)^2.*Exp[-(x^2) - (y + 1)^2] - 

   10*(x/5 - x^3 - y^5) Exp[-x^2 - y^2] - 1/3 Exp[-(x + 1)^2 - y^2];
N[Maximize[peaks[xx, yy], {xx, yy}]]
NMaximize[{peaks[x, y], x >= -2, x <= 2, y >= -2, y <= 3}, {x, y}]
peaks[0, 2]
peaks[1.28568, -0.00484756]

When you look at these graphs, you realize that the results are strange:

peaks[x_, y_] := 3*(1 - x)^2*Exp[-(x^2) - (y + 1)^2] - 
                 10*(x/5 - x^3 - y^5) Exp[-x^2 - y^2] - 1/3 Exp[-(x + 1)^2 - y^2];
Plot3D[peaks[x, y], {x, -2, 2}, {y, -2, 3}, 

    ColorFunction -> "DarkRainbow", 
    AxesLabel -> Automatic]
ContourPlot[peaks[x, y], {x, -2, 2}, {y, -2, 3},  
    ColorFunction -> "DarkRainbow", 
    ContourLines -> False,  
    Contours -> 25]

Mathematica graphics

Answer

You should use FindMaximum:

FindMaximum[peaks[xx, yy], {xx, yy}]

{8.106, {xx -> -0.009, yy -> 1.581}}

plot = Plot3D[peaks[x, y], {x, -2, 2}, {y, -2, 2}, PlotRange -> All];

point = Graphics3D[{PointSize@0.05, Point[{-0.009, 1.581, 8.106}]}];

Show[plot, point]

enter image description here

On the other hand, Maximize returns in this case the last local maximum it can find:

enter image description here

Comments

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