3.1. USING DERIVATIVES TO IDENTIFY EXTREME VALUES
167
Figure 3.5: Four possible graphs of a function f with a horizontal tangent line at a critical
point.
second derivative test.
Second Derivative Test: If p is a critical number of a continuous function f such
that f ′ (p) = 0 and f ′′ (p) 0, then f has a relative maximum at p if and only if
f ′′ (p) < 0, and f has a relative minimum at p if and only if f ′′ (p) > 0.
In the event that f ′′ (p) = 0, the second derivative test is inconclusive. That is, the test
doesn’t provide us any information. This is because if f ′′ (p) = 0, it is possible that f has
a local minimum, local maximum, or neither. 1
Just as a first derivative sign chart reveals all of the increasing and decreasing behavior
of a function, we can construct a second derivative sign chart that demonstrates all of the
important information involving concavity.
Example 3.2. Let f (x) be a function whose first derivative is f ′ (x) = 3x 4 −9x 2 . Construct
both first and second derivative sign charts for f , fully discuss where f is increasing and
decreasing and concave up and concave down, identify all relative extreme values, and
sketch a possible graph of f .
Solution. Since we know f ′ (x) = 3x 4 − 9x 2 , we can find the critical numbers of f by
solving 3x 4 − 9x 2 = 0. Factoring, we observe that
0 = 3x
2 (x
2 − 3) = 3x
2 (x +
√
3)(x −
√
3),
so that x = 0, ±
√
3 are the three critical numbers of f . It then follows that the first
derivative sign chart for f is given in Figure 3.6. Thus, f is increasing on the intervals
(−∞, −
√
3) and (
√
3, ∞), while f is decreasing on (−
√
3, 0) and (0,
√
3). Note particularly
that by the first derivative test, this information tells us that f has a local maximum at
1 Consider the functions f (x) = x 4 , g(x) = −x 4 , and h(x) = x 3 at the critical point p = 0.
167
Figure 3.5: Four possible graphs of a function f with a horizontal tangent line at a critical
point.
second derivative test.
Second Derivative Test: If p is a critical number of a continuous function f such
that f ′ (p) = 0 and f ′′ (p) 0, then f has a relative maximum at p if and only if
f ′′ (p) < 0, and f has a relative minimum at p if and only if f ′′ (p) > 0.
In the event that f ′′ (p) = 0, the second derivative test is inconclusive. That is, the test
doesn’t provide us any information. This is because if f ′′ (p) = 0, it is possible that f has
a local minimum, local maximum, or neither. 1
Just as a first derivative sign chart reveals all of the increasing and decreasing behavior
of a function, we can construct a second derivative sign chart that demonstrates all of the
important information involving concavity.
Example 3.2. Let f (x) be a function whose first derivative is f ′ (x) = 3x 4 −9x 2 . Construct
both first and second derivative sign charts for f , fully discuss where f is increasing and
decreasing and concave up and concave down, identify all relative extreme values, and
sketch a possible graph of f .
Solution. Since we know f ′ (x) = 3x 4 − 9x 2 , we can find the critical numbers of f by
solving 3x 4 − 9x 2 = 0. Factoring, we observe that
0 = 3x
2 (x
2 − 3) = 3x
2 (x +
√
3)(x −
√
3),
so that x = 0, ±
√
3 are the three critical numbers of f . It then follows that the first
derivative sign chart for f is given in Figure 3.6. Thus, f is increasing on the intervals
(−∞, −
√
3) and (
√
3, ∞), while f is decreasing on (−
√
3, 0) and (0,
√
3). Note particularly
that by the first derivative test, this information tells us that f has a local maximum at
1 Consider the functions f (x) = x 4 , g(x) = −x 4 , and h(x) = x 3 at the critical point p = 0.