Page 87 - How To Solve Word Problems In Calculus

Because it is impossible to find an open interval containing
0 for which (0, 0) is the highest point, f does not have a
relative maximum at 0. Similarly, it is impossible to find an
open interval containing 0 for which (0, 0) is the lowest point,
so f does not have a relative minimum at 0. f has no relative
extremum.
To decide whether a critical number is a relative maxi-
mum or relative minimum (or perhaps neither) we introduce
two tests called the first derivative test and the second derivative
test.

First Derivative Test

Let c be a critical value of f

(a)If f (x) changes from positive to negative as x goes
from the left of c to the right of c, then f has a relative
maximum at c.
(b)If f (x) changes from negative to positive as x goes

from the left of c to the right of c, then f has a relative
minimum at c.

Second Derivative Test

Let c be a critical value of f
(a)If f (c) < 0, then f has a relative maximum at c.

(b)If f (c) > 0, then f has a relative minimum at c.

If f (c) = 0 the second derivative test fails and the first deriva-

tive test must be used. However, this is very rare in solving
word problems and the second derivative test is often the more
convenient test to use.
When we solve a maximum-minimum problem, we are
looking for the largest or smallest value a function can attain.
We are looking for the absolute maximum or minimum value.
Why bother looking for relative extrema? The answer is given
in the following theorem:

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