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A key application of calculus is in optimization: finding maximum and minimum values of a function, and which points realize these extrema.
Formally, the field of mathematical optimization is called mathematical programming, and calculus methods of optimization are basic forms of nonlinear programming. We will primarily discuss finite-dimensional optimization, illustrating with functions in 1 or 2 variables, and algebraically discussing n variables. We will also indicate some extensions to infinite-dimensional optimization, such as calculus of variations, which is a primary application of these methods in physics.
Basic techniques include the first and second derivative test, and their higher-dimensional generalizations.
A more advanced technique is Lagrange multipliers, and generalizations as Karush–Kuhn–Tucker conditions and Lagrange multipliers on Banach spaces.
Optimization, particularly via Lagrange multipliers, is particularly used in the following fields:
- Particularly the Lagrangian formulation of classical mechanics.
- Neoclassical economics
- Input points, output values
- Maxima, minima, extrema, optima
- Stationary point, critical point; stationary value, critical value
- Objective function
- Constraints – equality and inequality
- Especially sublevel sets
- Feasible region, whose points are candidate solutions
This tutorial presents an introduction to optimization problems that involve finding a maximum or a minimum value of an objective function subject to a constraint of the form .
Maximum and minimumEdit
Finding optimum values of the function without a constraint is a well known problem dealt with in calculus courses. One would normally use the gradient to find stationary points. Then check all stationary and boundary points to find optimum values.
has one stationary point at (0,0).
Second derivative testEdit
The Second derivative test determines the optimality of stationary point according to the following rules :
- If at point x then has a local minimum at x
- If at point x then has a local maximum at x
- If has negative and positive eigenvalues then x is a saddle point
- Otherwise the test is inconclusive
In the above example.
Therefore has a minimum at (0,0).
-  T.K. Moon and W.C. Stirling. Mathematical Methods and Algorithms for Signal Processing. Prentice Hall. 2000.