Friday, November 2, 2012

Local extrema of continuous functions

Reversible continuous functions such as the increment function and the cube function have no extrema and therefore they are either monotone increasing or monotone decreasing on their entire domain. The square function has a single extrema at the origin because it is monotone increasing on the positive numbers and monotone decreasing on the negative numbers.

Local extrema can be determined by the first derivative of a function. For example, polynomials with positive coefficients always have derivative with positive coefficients so they can never have any positive extrema. For example, the exp function we has coefficients that are all positive reciprocals of the factorial numbers, is always reversible on the positive real numbers by the ln function.

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