Each point (x, y) in the plane is naturally assigned an “altitude” z, a function of x and y, also denoted by f (x, y). Plotting the points with coordinates (x, y, f (x, y)) in three dimensions produces something that may resemble a ski slope.
Graph of the function x ↦ f ( x, y ) =x 4 + y 4 – 2( x – y ) 2.
In the example shown, there is clearly a point somewhere that lies below all the others. That is the point our skier, tired after a long day on the slopes, would like to find. Mathematically, the task is to determine the values of x and y for which the altitude, f (x, y), is minimal. How can we locate it?
The art of approximation
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