The Applications of the Second Derivative
LRT
03/08/2017
Initial Remarks
The second derivative is the derivative of the first
derivative so it gives information on where the first
derivative is increasing, where it is decreasing, where it has
local maxima and where it has local minima.
If s(t) is the distance of an object travelling along a line
ds
from a fixed point, s0 (t) =
= v(t) is the velocity and
dt
00
0
s (t) = v (t) = a(t) is the acceleration.
Concavity
Let y = mx + b be any line with a slope (equivalently not a
vertical line).
This line divides the Cartesian plane into two subsets, the
points above the line and the points below the line.
above
below
Now let y = f (x), pick a point x = a and consider how the
curve looks with respect to the tangent line.
Look at either the red or the
green circle. You are seeing
the graph of a curve with a
tangent line at a point near
the center of the circle.
Notice that inside the red circle, the curve lies below the
tangent line.
Notice that inside the green circle, the curve lies above the
tangent line.
In the red case, we say the curve is concave down at the
point of tangency, x = a.
In the green case, we say the curve is concave up at the
point of tangency, x = a.
Concavity and the second derivative
The relation of the geometric discussion on the last slide
and the second derivative is the following.
If f 00 (a) > 0 then
the curve is concave up at the
point a, f (a) .
If f 00 (a) < 0 then
the curve is concave down at the
point a, f (a) .
On any interval (x0 , x1 ) where f 00 is continuous and f 00 > 0
and for any point a ∈ (x0 , x1 ), the tangent line to y = f (x)
at x = a lies above the curve on the entire interval.
On any interval (x0 , x1 ) where f 00 is continuous and f 00 < 0
and for any point a ∈ (x0 , x1 ), the tangent line to y = f (x)
at x = a lies below the curve on the entire interval.
Inflection Points
A point x = a on a graph y = f (x) is a point where a
tangent line exists (f 0 (a) or perhaps a vertical tangent) and
the concavity changes sign is called an inflection point.
Except for vertical tangents (which we can ignore for now),
an inflection point is a critical point of f 0 which is not a
relative extrema.
The Second Derivative Test
Suppose x = a is a critical point of y = f (x). The Second
Derivative Test can (sometimes) be used to decide if a is a
local maximum or a local minimum.
I
I
I
f 00 (a) > 0 implies a is a local minimum.
f 00 (a) < 0 implies a is a local maximum.
f 00 (a) = 0 implies nothing.
a can be a local maximum, or a local minimum, or
neither.
Examples
Find the intervals on which f is concave up or concave
down for the function f (x) = x4 − 2x3 + 6
f 0 (x) = 4x3 − 2 · 3x2 = 4x3 − 6x2
f 00 (x) = 4 · 3x2 − 6 · 2x = 12x2 − 12x
f 00 (x) = 12x2 − 12x = 12x(x − 1) = 0.
Hence x = 0 and x = 1 are the possible points where f 00
could change sign.
-
+
•
−1
+
•
0
0.5
•
1
2
To look at the Second Derivative Test, first find the critical
points
f 0 (x) = 4x3 − 6x2 = 2x2 (2x − 3) = 0 which has solutions
3
x = 0, x = .
2
Here are the two tangent
lines at the critical points.
Since f 0 = 0 at critical
points, tangent lines are horizontal. Notice at x = 0
the concavity switches sign
so an inflection point occurs
at x = 0. Since f 00 (0) =
0, this is an example of the
third possibility in the Second Derivative Test.
Here are the two tangent
lines at the critical points.
Since f 0 = 0 at critical
points, tangent lines are horizontal. Notice at x = 0
the concavity switches sign
so an inflection point occurs
at x = 0. Since f 00 (0) =
0, this is an example of the
third possibility in the Second Derivative Test.
The x = 1.5, f 00 (1.5) > 0 and the curve is concave up. The
Second Derivative Test says that the point x = 1.5 is a
local minimum.
Here is the graph of y =
x4 − 3x2 again with the
tangent lines at the two
inflection points. Notice
at the red one the concavity goes from up to
down and at the green
one it goes from down to
up.
Notice that between the two inflection points both tangent
lines lie above the curve. To the left of the red inflection
point the curve always lies above the tangent line. To the
right of green inflection point the curve always lies above
the tangent line.
The size of the second derivative measures how good
differential approximation works.
#84 continued
a. Drug-related crimes go up with or without the budget
cuts.
b. Drug-related crimes do not go up as fast if the budgets
aren’t cut.
#98
N (t) = 6.08t3 − 26.79t2 + 53.06t + 69.5
N 0 (t) = 18.24t2 − 53.58t + 53.06.
b2 − 4ac = −1000.4412 < 0 and N 0 (0) = 53.06 > 0 so N (t)
is increasing from (−∞, ∞).
b. askes, when is N 0 (t) as small as it gets on [0, 4]?A good
place to look for this is at a local minimum (for N 0 ) or
equivalently at an inflection point for N 00 .
N 00 (t) = 36.48t − 53.58 = 0.
So t ≈ 1.46 is this point and since N 00 is a line with positive
slope, it changes sign here from − to + and so t ≈ 1.46
produces a local minimum for N 0 . Since N 0 (1.46) ≈ 13.71
or about 14 communities a year.