Maximum and Minimum Values

The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.

Learning Objective

Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

Key Points

For a function of two variables, the second partial derivative test is based on the sign of $M(x,y)= f_{xx}(x,y)f_{yy}(x,y) - \left( f_{xy}(x,y) \right)^2$ and $f_{xx}(a,b)$, where $(a,b)$ is a critical point.
There are substantial differences between the functions of one variable and the functions of more than one variable in the identification of global extrema.
The maximum and minimum of a function, known collectively as extrema, are the largest and smallest values that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).

Terms

intermediate value theorem
a statement that claims that, for each value between the least upper bound and greatest lower bound of the image of a continuous function, there is a corresponding point in its domain that the function maps to that value
Rolle's theorem
a theorem stating that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero
critical point
a maximum, minimum, or point of inflection on a curve; a point at which the derivative of a function is zero or undefined

Full Text

The maximum and minimum of a function, known collectively as extrema, are the largest and smallest values that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).

Finding Maxima and Minima of Multivariable Functions

The second partial derivative test is a method in multivariable calculus used to determine whether a critical point $(a,b, \cdots )$ of a function $f(x,y, \cdots )$ is a local minimum, maximum, or saddle point.

Saddle Point

A saddle point on the graph of $z=x^2−y^2$ (in red).

For a function of two variables, suppose that $M(x,y)= f_{xx}(x,y)f_{yy}(x,y) - \left( f_{xy}(x,y) \right)^2$.

If $M(a,b)>0$ and $f_{xx}(a,b)>0$, then $(a,b)$ is a local minimum of $f$.
If M(a,b)>0M(a,b)>0 and fxx(a,b)<0f_{xx}(a,b)<0, then $(a,b)$ is a local maximum of $f$.
If $M(a,b)<0$, then $(a,b)$ is a saddle point of $f$.
If $M(a,b)=0$, then the second derivative test is inconclusive.

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). In two and more dimensions, this argument fails, as the function $f(x,y)= x^2+y^2(1-x)^3,\,\, x,y\in\mathbb{R}$ shows. Its only critical point is at $(0,0)$, which is a local minimum with $f(0,0) = 0$. However, it cannot be a global one, because $f(4,1) = 11$.

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