Minimum of a Function

For any function $ f $ defined on an interval $ I $ and $ m $ a real number belonging to $ I $, if $ f(x) <= f(m) $ on the interval $ I $ then $ f $ reaches its minimum in $ x=m $ over $ I $. In that case, $ f(m) $ is the minimum value of the function, reached when $ x=m $.

Finding the minimum of a function $ f $, is equivalent to calculate $ f(m) $. To find $ m $, use the derivative of the function. The minimum value of a function is found when its derivative is null and changes of sign, from negative to positive.

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

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

The minorant is any value lower than or equal to the minimum value reached by the function.

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

An line/affine function $ f (x) = ax + b $ always has for minimum $ -\infty $

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

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

A quadratic polynomial function of the form $ f(x) = ax^2+bx+c $ then

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

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