Page:Calculus Made Easy.pdf/136

 (3) Find the maxima and minima of $$y=\dfrac{x-1}{x^2+2}$$.

or $$x^2-2x-2=0$$, whose solutions are $$x =+2.73$$ and $$x=-0.73$$.

The denominator is always positive, so it is sufficient to ascertain the sign of the numerator.

If we put $$x=2.73$$, the numerator is negative; the maximum, $$y=0.183$$.

If we put $$x=-0.73$$, the numerator is positive; the minimum, $$y=-0.683$$.

(4) The expense $$C$$ of handling the products of a certain factory varies with the weekly output $$P$$ according to the relation $$C = aP + \dfrac{b}{c+P} + d$$, where $$a$$, $$b$$, $$c$$, $$d$$ are positive constants. For what output will the expense be least?

$$\dfrac{dC}{dP}=a-\frac{b}{(c+P)^2}=0\quad $$ for maximum or minimum;

hence $$a=\dfrac{b}{(c+P)^2}$$ and $$P = \pm \sqrt{\dfrac{b}{a}}-c$$.

As the output cannot be negative, $$P=+\sqrt{\dfrac{b}{a}}-c$$.