A likely answer given by a student of calculus would be, "It usually shows that a function reached a maximum or minimum. But it is not always so."

Just a
moment! The derivative f'(x) is the rate
of change of the value of function relative to the change of x. So f'(x_{0})
= 0 means that function f(x) is *almost
constant *around the value x_{0}.

In order to see that it is really true, do the following test:

Enter

\Y1=X^{2}

\Y2=X^{3}

\Y3=X^4

\Y4=X^5

Set the 2^{nd}
FORMAT to CoordOn, AxesOff,
and WINDOW to

Xmin=-.006

Xmax=.006

Ymin=-.004

Ymax=.004

GRAPH and use TRACE to see what is going on.

All these functions are almost constant around 0, which is the value where their derivatives are 0.

So what is the
connection between a function having a maximum at x_{0}, and being
almost constant around it?

Such a connection exists only for functions which have derivatives. Having a derivative means that a function can change only gradually. When the rate of change of such a function switches from positive to negative, there is always an in-between region where the function is almost constant. Functions without derivatives can have maxima that look like very sharp peaks, with no "almost constant" region around them.

Isaac Newton used functions that have derivatives to describe the movement of celestial bodies such as planets, moons, and comets. His second law, which states that the change of velocity of an object is proportional to the strength of the forces acting on it, tells us that the functions describing such motions can change their values only gradually. His discovery established the use of derivatives as a main and most powerful mathematical tool of the physical sciences.

Outside of astronomy most physical phenomena are not smooth, gradually changing, and orderly. Many objects have fractal structure and their changes can be sudden, explosive and turbulent. Functions describing such phenomena simply do not have derivatives.

But exact models of physical events are not always better, because in order to derive conclusions from a model we need adequate mathematical tools. Differential calculus provides such tools for functions having derivatives, but we do not have anything as powerful for more general classes of functions (for example, for arbitrary continuous functions). But there is one mathematical theorem that justifies the use of calculus for phenomena that cannot be described by functions having derivatives.

*Every continuous function can be
approximated with any required accuracy by a function which has a derivative.*

Thus in practice we use calculus as a mathematical modeling tool
for physical phenomena, not because it provides the best descriptions, but
because its descriptions are most useful. So the connection between a
function having a maximum at x_{0} and being almost constant around it,
is not a "fact of physics" that tells us something about the world, but an
artifact of the mathematical models we use to describe it.

Calculus Index