Page:Calculus Made Easy.pdf/139

(4) Find the maxima and minima of

$y=2x+1+{\frac {5}{x^{2}}}$.

(5) Find the maxima and minima of

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

(6) Find the maxima and minima of

$y={\frac {5x}{2+x^{2}}}$.

(7) Find the maxima and minima of

$y={\frac {3x}{x^{2}-3}}+{\frac {x}{2}}+5$.

(8) Divide a number $N$ into two parts in such a way that three times the square of one part plus twice the square of the other part shall be a minimum.

(9) The efficiency $u$ of an electric generator at different values of output $x$ is expressed by the general equation:

$u={\frac {x}{a+bx+cx^{2}}}$;

where $a$ is a constant depending chiefly on the energy losses in the iron and $c$ a constant depending chiefly on the resistance of the copper parts. Find an expression for that value of the output at which the efficiency will be a maximum.