Page:Calculus Made Easy.pdf/128

curve goes to the origin, as if there were a minimum there; but instead of continuing beyond, as it should do for a minimum, it retraces its steps (forming what is called a “cusp”). There is no minimum, therefore, although the condition for a minimum is satisfied, namely ${\dfrac {dy}{dx}}=0$. It is necessary therefore always to check by taking one value on either side.

Fig. 30.

Now, if we take $x={\tfrac {16}{25}}=0.64$. If $x=0.64$, $y=0.7373$ and $y=0.0819$; if $x=0.6$, $y$ becomes $0.6389$ and $0.0811$; and if $x=0.7$, $y$ becomes $0.8996$ and $0.0804$.

This shows that there are two branches of the curve; the upper one does not pass through a maximum, but the lower one does.

(7) A cylinder whose height is twice the radius of the base is increasing in volume, so that all its parts