Page:Calculus Made Easy.pdf/120

 Test Case.

Let us at once apply our knowledge to a case that we can test.

Let

and let us find whether this function has a maximum or minimum; and if so, test whether it is a maximum or a minimum.

Differentiating, we get

Equating to zero, we get

whence

or

That is to say, when $$x$$ is made $$=\tfrac{1}{2}$$, the corresponding value of $$y$$ will be either a maximum or a minimum. Accordingly, putting $$x=\tfrac{1}{2}$$ in the original equation, we get

or

Is this a maximum or a minimum? To test it, try putting $$x$$ a little bigger than $$\tfrac{1}{2}$$,–say make $$x=0.6$$. Then

which is higher up than $$-0.25$$; showing that $$y=-0.25$$ is a minimum.

Plot the curve for yourself, and verify the calculation.