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) =
3x^{2}  x^{3
}Then
f '(x) =
6x  3x^{2}
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) = x^{3}  x

f(x) = x^{4}  6x^{2
}

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
The Second Derivative Test
Let
f
be a twice differentiable function near
c
such that
f '(c) = 0. Then

If
f ''(x) > 0
then
f(c)
is a relative minimum.

If
f ''(x) < 0
then
f(c)
is a relative maximum.

If f ''(x) = 0 then use the first derivative test.

Example:
Let
f(x) = x^{4}  4x^{3
}Then
f '(x) = 4x^{3}  12x^{2}
which
is zero at x = 0 and x = 3
f ''(x) = 12x^{2}  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 + 3x^{2}  x^{3}

f(x) = x^{4 } 4x^{3} + 2

f(x) = ^{ }x + 4/x
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