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Minimum of a Function

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.

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Minimum of a Function -

Tag(s) : Mathematics

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# Minimum of a Function

## Minima Calculator

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.

### What is the definition of the minimum 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 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).

### How to find the minimum of a 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 $$\mathbb{R}$$.

### What is an extremum?

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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