Absolute Maxima
and Minima of Functions with Two Variables
In a Nut Shell: Absolute maximum and minimum
values of functions in the case of two independent variables, (x,y), occur at the "critical locations" within
or on the boundary of the domain, D. |
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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. You must investigate both
interior points and points on the boundary of the domain. |
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Locating critical points of a function, f(x,y
on the interior of the domain): 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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Strategy for finding Absolute Minimum, Absolute Maximum of f(x,y)
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