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Maximum of a Function

Tool to determine the maximum value of a function: the maximal value that can take a function. It is a global maximum and not a local maximum.

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Maximum of a Function -

Tag(s) : Mathematics

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# Maximum of a Function

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## Maxima Calculator

Tool to determine the maximum value of a function: the maximal value that can take a function. It is a global maximum and not a local maximum.

## Answers to Questions

### What is the definition of a maximum of a function?

For any function $$f$$ defined on an interval $$I$$ and $$m$$ a real of this interval, if $$f(x) <= f(m)$$ on the interval $$I$$ then $$f$$ reaches its maximum in $$x=m$$ over $$I$$. The value of the maximum is $$f(m)$$.

Example: $$f(x) = -x^2$$ defined over $$\mathbb{R}$$, the function reaches its maximum in $$x=0$$, $$f(x=0) = 0$$ and $$f(x) <= 0$$ over $$\mathbb{R}$$.

The maximum of a function is always defined with an interval, it can be local, or global : over the domain of definition of the function.

### How to calculate a maximum of a function?

The maximums of a function are detected when the derivative becomes null and changes its sign (passing through 0 from the positive side to the negative side).

Example: $$f(x) = -x^2 + 1$$ derivates as $$f'(x) = -2x$$ which is nullable in $$x = 0$$ as $$f'(x) = 0 \iff -2x = 0 \iff x = 0$$. An extremum is found in 0, its value is $$f(0) = 1$$. Calculations of limits $$\lim_{x\to0^-}{f'(x) = 0^+} \\ \lim_{x\to0^+}{f'(x) = 0^-}$$ show that the derivative change of sign from positive $$0^+$$ to negative $$0^-$$.

### What is an extremum?

An extremum is the name given to an extreme value of a function, a value that can be maximum (maximum of a function) or minimal (minimum of a function).

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