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The main purpose for determining critical points is to locate relative maxima and minima, as in single-variable calculus. When working with a function of one variable, the definition of a local extremum involves finding an interval around the critical point such that the function value is either greater than or less than all the other function values in that interval. When working with a function of two or more variables, we work with an open disk around the point.
Let be a function of two variables that is defined and continuous on an open set containing the point Then f has a local maximum at if
for all points within some disk centered at The number is called a local maximum value . If the preceding inequality holds for every point in the domain of then has a global maximum (also called an absolute maximum ) at
The function has a local minimum at if
for all points within some disk centered at The number is called a local minimum value . If the preceding inequality holds for every point in the domain of then has a global minimum (also called an absolute minimum ) at
If is either a local maximum or local minimum value, then it is called a local extremum (see the following figure).
In Maxima and Minima , we showed that extrema of functions of one variable occur at critical points. The same is true for functions of more than one variable, as stated in the following theorem.
Let be a function of two variables that is defined and continuous on an open set containing the point Suppose and each exists at If has a local extremum at then is a critical point of
Consider the function This function has a critical point at since However, does not have an extreme value at Therefore, the existence of a critical value at does not guarantee a local extremum at The same is true for a function of two or more variables. One way this can happen is at a saddle point . An example of a saddle point appears in the following figure.
In this graph, the origin is a saddle point. This is because the first partial derivatives of are both equal to zero at this point, but it is neither a maximum nor a minimum for the function. Furthermore the vertical trace corresponding to is (a parabola opening upward), but the vertical trace corresponding to is (a parabola opening downward). Therefore, it is both a global maximum for one trace and a global minimum for another.
Given the function the point is a saddle point if both and but does not have a local extremum at
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