Find the absolute maximum of f(x, y) = e^(-x^2-y^2)(x^2+2y^2) on x^2 + y^2 < 2

Notice that converting to polar coordinates, setting




allows us to consider as a function of one variable; let's call it , where



Then



We have for , and for , which means is increasing, then decreasing as exceeds 1. This suggests that extrema occur for wherever , i.e. along the intersection of the cylinder and .

Computing the second derivative of and setting equal to 0 gives



as a possible point of inflection. We have for , and namely when , which means is concave downward around this point. This confirms that is a site of a maximum. Along this path, we have a maximum value of .

Next, to check for possible extrema along the border, we can parameterize by and , so that



and we can think of as a function a single variable, , where



In other words, is constant along its boundary , and this is smaller than the maximum we found before.

So to recap, the maximum value of is , which is attained along the surface above the circle in the plane.

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