Objectives
After completing this chapter, you should be able to:
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Understand what it means for a function of two variables to have a local maximum, a local minimum, or a saddle point, and visualize each on the surface \(z = f(x,y)\text{.}\)
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Recognize critical points as the candidates for local extrema, and understand why a vanishing gradient is necessary but not sufficient for an extremum.
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Understand the idea behind the Second Derivative Test and what it reveals about the local shape of a surface near a critical point.
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Distinguish local extrema from absolute extrema, and understand why the absolute extrema of a function on a closed, bounded region may occur on the boundary.
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Interpret extreme values in a physical setting, such as the equilibrium and stability of a system.
