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) : Functions
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Minimum of a Function
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 $
Source code
dCode retains ownership of the "Minimum of a Function" source code. Any algorithm for the "Minimum of a Function" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "Minimum of a Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) or any database download or API access for "Minimum of a Function" or any other element are not public (except explicit open source licence like Creative Commons). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
Cite dCode
The content of the page "Minimum of a Function" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source.
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link:
In a scientific article or book, the recommended bibliographic citation is: Minimum of a Function on dCode.fr [online website], retrieved on 2025-04-16,
- Minimum Calculator
- Maximum Calculator
- What is the minimum of a function? (Definition)
- How to find the minimum of a function?
- How to calculate a local minimum over an interval?
- What is an extremum?
- What is a minorant of a function?
- What is the minimum of a constant function?
- What is the minimum of an affine function?
- What is the minimum of a 2nd degree polynomial function?
