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.



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