Figure 9:
Convexity and concavity.
![\begin{figure}\centering
\mbox{
\subfigure[]{\epsfig{file=ConvexFunction.eps,hei...
...quad
\subfigure[]{\epsfig{file=ConcaveFunction.eps,height=5cm}}
}\end{figure}](img1144.png) |
Proposition 6.10.6
Let
be a function defined and differentiable, at least twice, on an interval
.
- (i)
- If for every
, the function
is convex on
;
- (ii)
- If for every
, the function
is concave on
.
Example 6.10.7
Take

. The function

is a polynomial function, thus it is differentiable over

. We have:
Study the sign of the second derivative:
-
, i.e.
is concave over the interval
;
-
, i.e.
is convex over the interval
.
The graph of

is displayed on Figure
10.
Figure 10:
The graph of a cubic polynomial.
 |
Definition 6.10.8 (See Figure
11)
Let

be a function, defined on an interval

, and let

be an interior point of

.
Suppose that convexity and concavity interchange at

. The we say that

is
a point of inflection of

.
Figure 11:
A point of inflexion.
 |
According to Proposition 10.6, if
is (at least) twice differentiable in a neighborhood of
, then
is an inflection point of
provided the following conditions are fulfilled:
-
;
- the sign of
changes at
.
Example 6.10.9
- If
, then
has an inflection point at 0.
- The sine function has an inflection point at each
.
- The cosine function has an inflection point at each
.
- The exponential function
has no inflection point.
If
is a point of inflection of a function
, then at the point
, the graph of
passes through its tangent (see Figure 11).
Noah Dana-Picard
2007-12-28