Concavity

1
2.4 Geometrical Application of Calculus
Concavity
The second derivative gives us information about
the curves shape.

1. f(x) gt 0 - curve is concave upward.

1. f(x) lt 0 - curve is concave downward.

2
2.4 Geometrical Application of Calculus
Inflection
The second derivative gives us information about
the curves shape.

1. f(x) 0 - point of horizontal inflection.

3
2.4 Geometrical Application of Calculus
Exploring Stationary Points.
Yes, points of inflection as concavity does
change.
f(x) - / 0 /
/ 0 / -
No points of inflection as concavity does not
change.
f(x) / 0 /
- / 0 / -
4
2.4 Geometrical Application of Calculus
Exploring stationary points.
1. Does the curve y x4 have a point of
inflection?
Stationary point when 12x2 0
f(x) x4
f(x) 4x3
i.e. x 0
f(0) 0
Stationary point _at_ (0, 0)
f(x) 12x2
Test Concavity
x -1 0 1
f(x) 12 0 12
Concavity does not change - Not Inflection point.
5
2.4 Geometrical Application of Calculus
Exploring stationary points.
2. Does the curve y x3 have a point of
inflection?
Stationary point when 6x 0
f(x) x3
f(x) 3x2
i.e. x 0
f(0) 0
f(x) 6x
Stationary point _at_ (0, 0)
Test Concavity
x -1 0 1
f(x) -6 0 6
Concavity does change - Inflection point.