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118

Calculus Made Easy

This is clearly for minimum, since

${\frac {d^{2}C}{dE^{2}}}=(n+1){\frac {nC_{l}}{m}}E^{-(n+2)}$ ,

which is positive for a positive value of $E$ .

For a particular type of $16$ candle-power lamps, $C_{l}=17$ pence, $C_{e}=5$ pence; and it was found that $m=10$ and $n=3.6$ .

$E={\sqrt[{4.6}]{\frac {1000\times 3.6\times 17}{10\times 16\times 5}}}=2.6$ watts per candle-power.

Exercises X. (You are advised to plot the graph of any numerical example.) (See p. 258 for the Answers.)

(1) Find the maxima and minima of

$y=x^{3}+x^{2}-10x+8$ .

(2) Given $y={\dfrac {b}{a}}x-cx^{2}$ , find expressions for ${\dfrac {dy}{dx}}$ , and for ${\dfrac {d^{2}y}{dx^{2}}}$ , also find the value of $x$ which makes $y$ a maximum or a minimum, and show whether it is maximum or minimum.

(3) Find how many maxima and how many minima there are in the curve, the equation to which is

$y=1-{\frac {x^{2}}{2}}+{\frac {x^{4}}{24}}$ ;

and how many in that of which the equation is

$y=1-{\frac {x^{2}}{2}}+{\frac {x^{4}}{24}}-{\frac {x^{6}}{720}}$ .

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