In mathematical analysis , the maxima and minima the respective plurals of maximum and minimum of a function , known collectively as extrema the plural of extremum , are the largest and smallest value of the function, either within a given range the local or relative extrema or on the entire domain of a function the global or absolute extrema. As defined in set theory , the maximum and minimum of a set are the greatest and least elements in the set, respectively.
Unbounded infinite sets, such as the set of real numbers , have no minimum or maximum.
Applications of the Derivative
The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function. Symbolically, this can be written as follows:. A similar definition can be used when X is a topological space , since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows:. In both the global and local cases, the concept of a strict extremum can be defined. Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points.
- How Do We Know it is a Maximum (or Minimum)?.
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A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed and bounded interval of real numbers see the graph above. Finding global maxima and minima is the goal of mathematical optimization.
If a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist.
Furthermore, a global maximum or minimum either must be a local maximum or minimum in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum or minimum is to look at all the local maxima or minima in the interior, and also look at the maxima or minima of the points on the boundary, and take the largest or smallest one.
Likely the most important, yet quite obvious, feature of continuous real-valued functions of a real variable is that they decrease before local minima and increase afterwards, likewise for maxima. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test , second derivative test , or higher-order derivative test , given sufficient differentiability.
For any function that is defined piecewise , one finds a maximum or minimum by finding the maximum or minimum of each piece separately, and then seeing which one is largest or smallest. For functions of more than one variable, similar conditions apply. For example, in the enlargeable figure at 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.
Maxima and minima
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 throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum.
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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 reductio ad absurdum.
Maximums and Minimums
In two and more dimensions, this argument fails, as the function. If the domain of a function for which an extremum is to be found consists itself of functions, i. Maxima and minima can also be defined for sets. In general, if an ordered set S has a greatest element m , m is a maximal element. Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with respect to order induced by T , m is a least upper bound of S in T.
Before we develop such a test, we do one more example that sheds more light on the issues our test needs to consider. A point that seems to act as both a max and a min is a saddle point. A formal definition follows. Before Example We now recognize that our test also needs to account for saddle points.
Extreme Values (Minimum/Maximum) Function Calculator - Online Tool
We called this the Second Derivative Test. Note that at a saddle point, it seems the graph is "both'' concave up and concave down, depending on which direction you are considering. However, this is not the case. We first practice using this test with the function in the previous example, where we visually determined we had a relative maximum and a saddle point.
Determine whether the function has a relative minimum, maximum, or saddle point at each critical point. The Second Derivative Test confirmed what we determined visually. Note how this function does not vary much near the critical points -- that is, visually it is difficult to determine whether a point is a saddle point or relative minimum or even a critical point at all!
This is one reason why the Second Derivative Test is so important to have. A similar theorem and procedure applies to functions of two variables. A continuous function over a closed set also attains a maximum and minimum value see the following theorem. We can find these values by evaluating the function at the critical values in the set and over the boundary of the set. After formally stating this extreme value theorem, we give examples. In Figure We checked each vertex of the triangle twice, as each showed up as the endpoint of an interval twice.
This portion of the text is entitled "Constrained Optimization'' because we want to optimize a function i. In the previous example, we constrained ourselves by considering a function only within the boundary of a triangle.