## You are here

Homeleast and greatest value of function

## Primary tabs

# least and greatest value of function

###### Theorem.

If the real function $f$ is

1. continuous on the closed interval $[a,\,b]$ and

2. differentiable on the open interval $(a,\,b)$,

then the function has on the interval $[a,\,b]$ a least value and a greatest value. These are always got in the end of the interval or in the zero of the derivative.

Remark 1. If the preconditions of the theorem are fulfilled by a function $f$, then one needs only to determine the values of $f$ in the end points $a$ and $b$ of the interval and in the zeros of the derivative $f^{{\prime}}$ inside the interval; then the least and the greatest value are found among those values.

Remark 2. Note that the theorem does not require anything of the derivative $f^{{\prime}}$ in the points $a$ and $b$; one needs not even the right-sided derivative in $a$ or the left-sided derivative in $b$. Thus e.g. the function $f:\,x\mapsto\sqrt{1-x^{2}}$, fulfilling the conditions of the theorem on the interval $[-1,\,1]$ but not having such one-sided derivatives, gains its least value in the end-point $x=-1$ and its greatest value in the zero $x=0$ of the derivative.

Remark 3. The least value of a function is also called the *absolute minimum* and the greatest value the *absolute maximum* of the function.

## Mathematics Subject Classification

26B12*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections