Page:Calculus Made Easy.pdf/114

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

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$y$	$7$	$4$	$3$	$4$	$7$	$12$

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

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.

Fig. 27.

Here is another example:

Let

$y=3x-x^{2}$.

Calculate a few values thus:

$x$	$-1$	$0$	$1$	$2$	$3$	$4$	$5$
$y$	$-4$	$0$	$2$	$2$	$0$	$-4$	$-10$