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.

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 \( 0^- \).

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).