Convexity and concavity of a function.

Definition 6.10.5   Let $ f$ be a function defined over an interval $ I$ .
(i)
The function $ f$ is convex on $ I$ if, for any two points $ A$ and $ B$ on the graph, the arc $ AB$ is below the chord $ AB$ (see Figure 9(a)).
(ii)
The function $ f$ is concave on $ I$ if, for any two points $ A$ and $ B$ on the graph, the chord $ AB$ is below the arc $ AB$ (see Figure 9(b)).

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}

Proposition 6.10.6   Let $ f$ be a function defined and differentiable, at least twice, on an interval $ I$ .
(i)
If for every $ x \in I, \; f''(x)>0$ , the function $ f$ is convex on $ I$ ;
(ii)
If for every $ x \in I, \; f''(x)<0$ , the function $ f$ is concave on $ I$ .

Example 6.10.7   Take $ f(x)=x^3-3x^2+1$ . The function $ f$ is a polynomial function, thus it is differentiable over $ \mathbb{R}$ . We have:

$\displaystyle \forall x \in \mathbb{R}, \;$ $\displaystyle f'(x) = 3x^2-6x$    
$\displaystyle \quad$ $\displaystyle f''(x)=6x-6.$    

Study the sign of the second derivative: The graph of $ f$ is displayed on Figure 10.
Figure 10: The graph of a cubic polynomial.
\begin{figure}\centering
\mbox{\epsfig{file=CubicPolynomial.eps,height=6cm}}\end{figure}

Definition 6.10.8 (See Figure 11)   Let $ f$ be a function, defined on an interval $ I$ , and let $ x_0$ be an interior point of $ I$ . Suppose that convexity and concavity interchange at $ x_0$ . The we say that $ x_0$ is a point of inflection of $ f$ .

Figure 11: A point of inflexion.
\begin{figure}\mbox{\epsfig{file=Inflection.eps,height=4cm}}\end{figure}

According to Proposition 10.6, if $ f$ is (at least) twice differentiable in a neighborhood of $ x_0$ , then $ x_0$ is an inflection point of $ f$ provided the following conditions are fulfilled:

Example 6.10.9  
  1. If $ f(x)=x^3$ , then $ f$ has an inflection point at 0.
  2. The sine function has an inflection point at each $ k \pi$ .
  3. The cosine function has an inflection point at each $ \frac {\pi}{2}+ k \pi$ .
  4. The exponential function $ x \mapsto e^x$ has no inflection point.

Remark 6.10.10   Denote by $ \mathcal{C}$ the graph of the function $ f$ and by $ M_0$ the point whose coordinate are $ (x_0, f(x_0)$ .

See Figure 12.
Figure 12: Tangent either on top or below.
\begin{figure}\mbox{\epsfig{file=ConvexConcave.eps,height=4cm}}\end{figure}

If $ x_0$ is a point of inflection of a function $ f$ , then at the point $ M_0(x_0, f(x_0))$ , the graph of $ f$ passes through its tangent (see Figure 11).

Noah Dana-Picard 2007-12-28