What is the definition of an extremum of a function?

Question

In calculus extremum is a common topic, but I don't understand what it is. What is the use of this extremum, especially in practical life?

Answer 1

EDIT: As to what an extrema is: Extrema are maximums and minimums--over its whole "lifetime" how high and how low does a function go?

Original post:

Many, many processes in "real life" can be represented by functions. These applications can range in topic from physics (for example, the velocity of an object), to financial (cost to produce a certain quantity of goods), to the mundane (how much water you use per day, on an average basis).

Finding the extreme values of the functions is done whenever knowing the max or min is necessary. For example--how high will my model rocket go? That's a calculus problem.

Another example would be to find the maximal profit you can earn with a certain production method. Or, what is the maximal volume of a box that uses a fixed amount of material (or even a minimal amount of material)? What is the minimum amount of fencing to be used for your cattle herd?

These are just some simple examples--there are thousands of applications out there, it is just a matter of finding a process and needing to maximize or minimize its output.

Answer 2

In analysis of differentiable real-valued functions of a single argument, an extremum is usually defined to be a place where the derivative is zero.

$$ x_0 \text{ is an extremum of } f(x) \text{ iff } f'(x_0) = 0 $$

To identify an extremum as a local maximum, minimum or an inflection point, you look at the second derivative:

$$ x_0 \text{ is a local maximum of } f(x) \text{ iff } f'(x_0)=0\text{ and } f''(x_0)<0 $$ $$ x_0 \text{ is a local minimum of } f(x) \text{ iff } f'(x_0)=0\text{ and } f''(x_0)>0 $$ $$ x_0 \text{ is an inflection point of } f(x) \text{ iff } f'(x_0)= f''(x_0) = 0 $$

Answer 3

An extremum (plural: extrema) is a maximum (plural: maxima) or a minimum.

Let $f$ be a function on $I\subset\mathbb{R}$ and let $x_0\in\ I$. We say that $f$ have a local maximum (resp. local minimum) at $x_0$ if $f(x)<f(x_0)$ (resp. $f(x)>f(x_0)$) in the strict neighborhood of $x_0$ (i.e. $|x-x_0|<\epsilon$ for some $\epsilon>0$ and $x\not=x_0$).

$f$ have a global maximum (resp. global minimum) at $x_0$ if $f(x)<f(x_0)$ (resp. $f(x)>f(x_0)$) for all $x\in I$.

The definitions are clear but the problem of finding $x_0$ directly from the definition is often very difficult so we usually use the techniques of calculus (the use of derivatives).

Note: if $I$ is compact then $f$ has a global maximum and a global minimum.