Maximum of a Function

A function maximum is the point where the function reaches its greatest value. Formally, for any function $ f(x) $ 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) $.

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).

Add one or more constraints indicating the conditions for each variable.

A local maximum is the highest point in a neighborhood/interval, while a global maximum is the highest point over the entire domain of the function.

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).

The majorant is any value greater than or equal to the maximum value reached by the function.

A constant function $ f (x) = c $ is a line, and always equals $ c $, so its maximum is $ c $, reached for any value of $ x $

An affine function $ f (x) = ax + b $ is a line that always has for maximum $ +\infty $

— If $ a < 0 $, the maximum of $ f $ is $ +\infty $ when $ x $ tends to $ -\infty $

— If $ a > 0 $, the maximum of $ f $ is $ +\infty $ when $ x $ tends to $ +\infty $

For a quadratic polynomial function $ f(x) = ax^2 + bx + c $ then

— If $ a < 0 $, the maximum of $ f $ is $ (-b^2 + 4 a c)/(4 a) $ reached when $ x = -\frac{b}{2a} $

— If $ a > 0 $, the maximum of $ f $ is $ +\infty $ when $ x $ tends to $ +\infty $