Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/193

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IV. Let and .
By continuing the process as illustrated in I, II, and III, it is seen that if the first derivative of which does not vanish for x = a is of even order (= n), then
(47) is a maximum if a negative number;
(48) is a minimum if a positive number.[1]
If the first derivative of which does not vanish for x = a is of odd order, then will be neither a maximum nor a minimum.

Illustrative Example 1. Examine for maximum and minimum values.

Solution. .
  .
Solving
gives the critical values x = 2 and x = 4. ∴ , and .
Differentiating again, .
Since , we know from (47) that is a maximum.
Since , we know from (48) that is a minimum.

Illustrative Example 2. Examine for maximum and minimum values.

Solution. ,
  , for x = 0,[2]
  , for x = 0,
  , for x = 0,
  , for x = 0.
Hence from (48), is a minimum.
EXAMPLES

Examine the following functions for maximum and minimum values, using the method of the last section:

1. . Ans. x = 1 gives min. = 0;
  x = 0 gives neither.
2. .   x = 2 gives neither.
3. .   x = 1 gives min. = 0 ;
  gives max.;
  x = -1 gives neither.

4. Investigate , at x = 1 and x = 3.

5. Investigate , at x = 1.

6. Show that if the first derivative of which does not vanish for x = a is of odd order (= n), then is an increasing or decreasing function when x = a, according as is positive or negative.

  1. As in § 82, a critical value x = a is found by placing the first derivative equal to zero and solving the resulting equation for real roots.
  2. x = 0 is the only root of the equation .