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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 \( f(m) \).

Example: 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 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).

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 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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