Tool to determine the minimum value of a function: the minimal value that can take a function. It is a global minimum and not a local minimum.

Minimum of a Function - dCode

Tag(s) : Mathematics

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Tool to determine the minimum value of a function: the minimal value that can take a function. It is a global minimum and not a local minimum.

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 minimum in \( x=m \) over \( I \). In that case, \( f(m) \) is the minimum value of the function, reached when \( x=m \).

The **minimum of a function** is always defined with an interval (which may be the domain of definition of the function).

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.

Example: \( f(x) = x^2 \) defined over \( \mathbb{R} \), its derivative is \( f'(x) = 2x \), that is equal to zero in \( x = 0 \) because \( f'(x) = 0 \iff 2x = 0 \iff x=0 \). The derivative goes from negative to positive in \( x = 0 \) so the function has a minimum in \( x=0 \), \( f(x=0) = 0 \) and \( f(x) >= 0 \) over \( \mathbb{R} \).

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).

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