100
Calculus Made Easy
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 is made , the corresponding value of will be either a maximum or a minimum. Accordingly, putting in the original equation, we get
,
or
.
Is this a maximum or a minimum? To test it, try putting a little bigger than ,–say make . Then
,
which is higher up than ; showing that is a minimum.
Plot the curve for yourself, and verify the calculation.