Maximum of a Function

For any function $ f $ defined on an interval $ I $, taking $ m $ a real of this interval, if $ f(x) <= f(m) $ over the whole 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 (between 2 values), 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: Calculate the maximum of the function $ f(x) = -x^2 + 1 $. This function has for derivative $ 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^- $. Global extremun of the function is then $ 1 $ for $ x=0 $.

dCode has also a minimum of a function calculator tool.

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).