Critical Points

Critical points are points where the derivative of a function is equal to zero.

Single Variable

Multivariable

In multivariable calculus critical points are points where the partial derivatives of the function both equal zero.

To find the critical points of a function one finds the partial derivatives of the function, sets them equal to zero, and then solves the resulting system of equations.

Once the critical points have been obtained the second derivative test can be used to define each point as a minimum, maximum, or a saddle point.

Over a Defined Region

If your function is constrained within a region then your function must have an absolute minimum and maximum within that region.

The above theorem is called the extreme value theorem.

One can find these points via the following:

  1. Find the interior (inside the region) critical points of the function using the method specified in the section above.
  2. List the boundary points where the function might have local minimum and maximum values and evaluate the function at each of those points
  3. The largest points from steps 1 and 2 are the absolute maximum points, the smallest are the absolute minimum points.

See Multivariable Calculus Notes - Chapter 14 - Sections 7-8 for more information and examples.

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Table Of Contents
Critical Points
Single Variable
Multivariable
Over a Defined Region