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96

Calculus Made Easy

Differentiating, we get:

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

Now equate this to zero, thus:

$2x-4=0$ .

Solving this equation for $x$ , we get:

$2x$	$=4$ ,
$x$	$=2$ .

Now, we know that the maximum (or minimum) will occur exactly when $x=2$ .

Putting the value $x=2$ into the original equation, we get

Now look back at Fig 26, and you will see that the minimum occurs when $x=2$ , and that this minimum of $y=3$ .

Try the second example (Fig. 24), which is

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

Differentiating,

${\frac {dy}{dx}}=3-2x$ .

Equating to zero,

$3-2x=0$ ,

whence

$x=1{\tfrac {1}{2}}$ ;

and putting this value of $x$ into the original equation, we find:

$y$	$=4{\tfrac {1}{2}}-(1{\tfrac {1}{2}}\times 1{\tfrac {1}{2}})$ ,
$y$	$=2{\tfrac {1}{4}}$ .

This gives us exactly the information as to which the method of trying a lot of values left us uncertain.