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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) : Functions

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Minimum of a Function

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Answers to Questions (FAQ)

What is the minimum of a function? (Definition)

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 $.

The minimum of a function is always defined with an interval (which may be the whole 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} $.

How to calculate a local minimum over an interval?

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

Example: Find the minimum of $ \sin{x} $ for $ 0 < x < \pi $

Indicate several equations with the operator logical AND && to separate the equations

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).

What is a minorant of a function?

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

What is the minimum of a constant 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 $

What is the minimum of an affine function?

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 $

What is the minimum of a 2nd degree polynomial function?

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 $

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Minimum of a Function on dCode.fr [online website], retrieved on 2022-08-08, https://www.dcode.fr/minimum-function

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