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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

How to calculate a maximum?

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 \( f(m) \).

Consider \( f(x) = -x^2 \) 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 found when it's derivative is null and changes of sign, from positive to negative.

The maximum of a function is always defined with an interval (which may be the domain of definition of the function).

What is an extremum?

An extremum is an extreme value of a function, this value can be maximum (one speaks of a maximum of a function) or minimum (one speaks of a minimum of a function).

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