Editing Maximum and minimum (section)

==Functions of more than one variable==<!-- This section is linked from [[Indifference curve]] -->
{{main|Second partial derivative test}}
[[File:Modell einer Peanoschen Fläche -Schilling XLIX, 1-.jpg|thumb|left|[[Peano surface]], a counterexample to some criteria of local maxima of the 19th century]][[File:MaximumParaboloid.png|thumb|right|The global maximum is the point at the top]]
[[File:MaximumCounterexample.png|thumb|right|Counterexample: The red dot shows a local minimum that is not a global minimum]]
For functions of more than one variable, similar conditions apply. For example, in the (enlargeable) figure on the right, the necessary conditions for a ''local'' maximum are similar to those of a function with only one variable. The first [[partial derivatives]] as to ''z'' (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure).  The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of the possibility of a [[saddle point]]. For use of these conditions to solve for a maximum, the function ''z'' must also be [[Differentiable function|differentiable]] throughout. The [[second partial derivative test]] can help classify the point as a relative maximum or relative minimum.
In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if a bounded differentiable function ''f'' defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the [[intermediate value theorem]] and [[Rolle's theorem]] to prove this by [[proof by contradiction|contradiction]]). In two and more dimensions, this argument fails. This is illustrated by the function
:<math>f(x,y)= x^2+y^2(1-x)^3,\qquad x,y \in \R,</math>
whose only critical point is at (0,0), which is a local minimum with ''f''(0,0)&nbsp;=&nbsp;0. However, it cannot be a global one, because ''f''(2,3)&nbsp;=&nbsp;−5.