readily see that the equation represents a curve with a minimum.

${\displaystyle x}$	${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle 3}$	${\displaystyle 4}$	${\displaystyle 5}$
${\displaystyle y}$	${\displaystyle 7}$	${\displaystyle 4}$	${\displaystyle 3}$	${\displaystyle 4}$	${\displaystyle 7}$	${\displaystyle 12}$

These values are plotted in Fig. 26, which shows that ${\displaystyle y}$ has apparently a minimum value of ${\displaystyle 3}$, when ${\displaystyle x}$ is made equal to ${\displaystyle 2}$. But are you sure that the minimum occurs at ${\displaystyle 2}$, and not at ${\displaystyle 2{\tfrac {1}{4}}}$ or at ${\displaystyle 1{\tfrac {3}{4}}}$?

Of course it would be possible with any algebraic expression to work out a lot of values, and in this way arrive gradually at the particular value that may be a maximum or a minimum.

Here is another example:

Let

Calculate a few values thus:

${\displaystyle x}$	${\displaystyle -1}$	${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle 3}$	${\displaystyle 4}$	${\displaystyle 5}$
${\displaystyle y}$	${\displaystyle -4}$	${\displaystyle 0}$	${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 0}$	${\displaystyle -4}$	${\displaystyle -10}$