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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 - dCode
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Minimum of a Function
- Mathematics
- Functions
- Minimum of a Function
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Answers to Questions (FAQ)
What is the definition of the 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 $.
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 $
What is an extremum?
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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how to find minimum value
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