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MAXIMA AND MINIMA

101

Further Examples.

A most interesting example is afforded by a curve that has both a maximum and a minimum. Its equation is:

$y={\tfrac {1}{3}}x^{3}-2x^{2}+3x+1$ .

Now

${\dfrac {dy}{dx}}=x^{2}-4x+3$ .

Fig. 28.

Equating to zero, we get the quadratic,

$x^{2}-4x+3=0$ ;

and solving the quadratic gives us two roots, viz.

${\begin{cases}x=3\\x=1.\end{cases}}$

Now, when $x=3$ , $y=1$ ; and when $x=1$ , $y=2{\tfrac {1}{3}}$ . The first of these is a minimum, the second a maximum.

The curve itself may be plotted (as in Fig. 28)

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