Page:Calculus Made Easy.pdf/138

This is clearly for minimum, since

which is positive for a positive value of ${\displaystyle E}$.

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

${\displaystyle E={\sqrt[{4.6}]{\frac {1000\times 3.6\times 17}{10\times 16\times 5}}}=2.6}$ 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

(2) Given ${\displaystyle y={\dfrac {b}{a}}x-cx^{2}}$, find expressions for ${\displaystyle {\dfrac {dy}{dx}}}$, and for ${\displaystyle {\dfrac {d^{2}y}{dx^{2}}}}$, also find the value of ${\displaystyle x}$ which makes ${\displaystyle y}$ 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

and how many in that of which the equation is