The Second Derivative Test
Concavity and Inflection Points
Example:
Consider a typical ski slope. At what point on
the slope are you having the most fun? In other words where is the
slope the greatest?
Definition If f '(x) is increasing, then the function is concave up and if f '(x) is decreasing then the function is concave down. To determine whether the derivative is increasing, we take the second derivative. |
Example:
Let
f(x) =
3x2 - x3
Then
f '(x) =
6x - 3x2
and
f ''(x) =
6 - 6x.
Solving
6 - 6x >
0
we see that the function is concave up when x <1.
Solving
6 - 6x <
0
we see that the function is concave down when x > 1.
When x = 1 we say that f(x) has an inflection point.
Exercise: Determine where the function is concave up and concave
down:
-
f(x) = x3 - x
-
f(x) = x4 - 6x2
-
The Second Derivative Test
Suppose that a twice differentiable function has a relative maximum at x
= c. The by the first derivative test, the derivative is positive to
the left of c and negative to the right of c. Going from positive to
negative means the derivative is decreasing at x = c. A decreasing
first derivative implies a negative second derivative. Similarly at a
relative minimum the second derivative is positive. This implies
|
Let f be a twice differentiable function near c such that f '(c) = 0. Then
|
Example:
Let
f(x) = x4 - 4x3
Then
f '(x) = 4x3 - 12x2
which
is zero at x = 0 and x = 3
f ''(x) = 12x2 - 24x
f ''(0) = 0
and
f ''(3) > 0
Hence at x = 3 there is a relative minimum. At x = 0 we use the first
derivative test. To the left of 0,
f '(x) < 0
and to the right of 0,
f '(x) <0
thus at x = 0 there is no local extrema.
Exercises:
For the following determine the concavity, inflection points, relative
extrema and intervals where the function is increasing and intervals where
the function is decreasing.
-
f(x) = 5 + 3x2 - x3
-
f(x) = x4 - 4x3 + 2
-
f(x) = x + 4/x