Local Maxima and
Minima of Functions with Two Variables
In a Nut Shell: Maximum and minimum values of
functions in the case of two independent variables, (x,y), are similar to the case with one independent
variable, x, in that they occur where
the slope at the "critical location" i.e.
(a,b) is zero. |
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A local maximum occurs near (a,b) if f(x,y) ≤ f(a,b). Likewise a local minimum occurs near (a,b) if f(x,y) ≥ f(a,b). An absolute maximum occurs at (a,b) if f(x,y) ≤ f(a,b) for all points (x,y) in the domain of the function and an absolute minimum occurs at (a,b)
if f(x,y) ≥ f(a,b)
for all points (x,y) in the domain of the function. It is possible that the
function, f(x,y), has neither a maximum nor a
minimum at a critical point, (a,b). In this case
point (a,b) is called a "saddle point". |
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Locating critical points of a function, f(x,y): The slope of the function,
f(x,y), must be zero at each critical point, (a,b). For
functions of two independent
variables, (x,y), the following partial derivatives
must hold: ∂f(a,b)/∂x
= 0 and ∂f(a,b)/∂y = 0 |
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Procedure for finding Local Minimum, Local Maximum, and Saddle Points
of f(x,y)
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Copyright © 2017 Richard C. Coddington
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