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'''Maxima''' and '''minima''' are a type of '''extrema''' or critical points (numbers) for functions, which can be found with calculus.

'''Maxima''' are a type of extrema or critical number for functions, which can be found with calculus.

The ''absolute'' maximum of a function is the point in the domain at which the function achieves its largest value.  For example, the absolute maximum of <math>f(x) = 1- x^2</math> occurs at <math>x=0</math>, where <math>f(0) = 1</math>.

The ''absolute'' maximum of a function is the point in the domain at which the function achieves its largest value.  For example, the absolute maximum of <math>f(x) = 1- x^2</math> occurs at <math>x=0</math>, where <math>f(0) = 1</math>.

To find the '''maxima''' of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero.

To find the '''maxima''' of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero.

The [[minima]] of a function can be calculated in an analogous manner.

The '''minima''' of a function can be calculated in an analogous manner.

The computation of maxima of functions remains an important issue in more advanced mathematics.  For example, it is often useful to consider real-valued continuous functions on more exotic topological spaces, and ask what their maxima are.  Note that some functions do not have maxima: for example, <math>f(x) = x^2</math> takes on arbitrarily large values.  A fundamental theorem states that any function defined on a [[compact space]] obtains a maximum value; this is just a consequence of the fact that the image of a compact space under a continuous function is compact, and thus bounded.

The computation of maxima of functions remains an important issue in more advanced mathematics.  For example, it is often useful to consider real-valued continuous functions on more exotic topological spaces, and ask what their maxima are.  Note that some functions do not have maxima: for example, <math>f(x) = x^2</math> takes on arbitrarily large values.  A fundamental theorem states that any function defined on a [[compact space]] obtains a maximum value; this is just a consequence of the fact that the image of a compact space under a continuous function is compact, and thus bounded.