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How do you find the minimum or maximum of a function and example?

Updated: 8/19/2019
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Hassanjan28

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12y ago

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It depends on the function.

Some functions, for example any polynomial of odd order, will have no maximum or minimum.

Some functions, such as the sine or cosine functions, will have an infinite number of maxima and minima.

If a function is differentiable then a turning point can be found by finding the zero of its derivative. This could be a maximum, minimum or a point of inflexion. If the derivative before this zero is negative and after the zero is positive then the point is a minimum. If it goes from positive to negative, the pont is a maximum, and if it has the same sign (either both +ve or both -ve) then it is a point of inflexion. A second derivative can help answer this quicker, but it need not exist.

These are all well behaved functions. The task is much more complicated for ill behaved functions. Consider, for example, the difference between consecutive primes. The minimum is clearly 1 (between 2 and 3) but the maximum? Or the number of digits between 1 and 4 in the decimal expansio of pi = 3.14159.... Minimum digit between = 0 (they are consecutive near the start of pi), but maximum?

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Q: How do you find the minimum or maximum of a function and example?
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