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.

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:

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

$$\quad f''(x)=6x-6.$$

Study the sign of the second derivative:

• $f''(x)<0 \Longleftrightarrow x<1$ , i.e. $f$ is concave over the interval $(-\infty, 1)$ ;
• $f''(x)>0 \Longleftrightarrow x>1$ , i.e. $f$ is convex over the interval $(1,+\infty)$ .

The graph of $f$ is displayed on Figure 10.

Figure 10: The graph of a cubic polynomial.

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.

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:

• $f''(x_0)=0$ ;
• the sign of $f''$ changes at $x_0$ .

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 $(x_0, f(x_0)$ .

• If the function is concave at a point $x_0$ , then the tangent to $\mathcal{C}$ at $M_0$ is over $\mathcal{C}$ in a neighborhood of $M_0$ .
• If the function is convex at a point $x_0$ , then the tangent to $\mathcal{C}$ at $M_0$ is under $\mathcal{C}$ in a neighborhood of $M_0$ .

See Figure 12.

Figure 12: Tangent either on top or below.

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).