#### Chapter 7 Differential Calculus

**Section 7.4 Properties of Functions**

# 7.4.3 Second Derivative and Bending Properties (Curvature)

Let us consider a function $f:D\to \mathbb{R}$ that is differentiable on the interval $\text{}]a;b[\text{}\subseteq D$. If its derivative $f\text{'}$ is also differentiable on the interval $\text{}]a;b[\text{}\subseteq D$, then $f$ is called

**twice-differentiable**. The derivative of the first derivative of $f$ ($(f\text{'})\text{'}=f\text{'}\text{'}$) is called the

**second derivative**of the function $f$.

The second derivative of the function $f$ can be used to investigate the bending behaviour (curvature) of the function:

##### **Bending Properties (Curvature) 7.4.2 **

If $f\text{'}\text{'}(x)\ge 0$ for all $x$ between $a$ and $b$, then $f$ is called

If $f\text{'}\text{'}(x)\le 0$ for all $x$ between $a$ and $b$, then $f$ is called

**convex**(**left curved**or**concave up**) on the interval $\text{}]a;b[\text{}$.If $f\text{'}\text{'}(x)\le 0$ for all $x$ between $a$ and $b$, then $f$ is called

**concave**(**right curved**or**concave down**) on the interval $\text{}]a;b[\text{}$.Thus, it is sufficient to determine the sign of the second derivative $f\text{'}\text{'}$ to decide whether a function is convex (left curved) or concave (right curved).

##### **Comment on the Notation 7.4.3 **

The second derivative and further "higher" derivatives are often denoted using superscript natural numbers in round brackets: ${f}^{(k)}$ then denotes the $k$th derivative of $f$. In particular, this notation is used in generally written formulas even for the (first) derivative ($k=1$) and for the function $f$ itself ($k=0$).

Hence,

Hence,

- ${f}^{(0)}=f$ denotes the function $f$,

- ${f}^{(1)}=f\text{'}$ denotes the (first) derivative,

- ${f}^{(2)}=f\text{'}\text{'}$ the second derivative,

- ${f}^{(3)}$ the third derivative, and

- ${f}^{(4)}$ the fourth derivative of $f$.

The following example shows that a monotonically increasing function can be convex on one region and concave on another.

##### **Example 7.4.4 **

Certainly, the function $f:\mathbb{R}\to \mathbb{R},x\to {x}^{3}$ is at least twice-differentiable. Since $f\text{'}(x)=3{x}^{2}\ge 0$ for all $x\in \mathbb{R}$, the function $f$ is monotonically increasing on its entire domain. Moreover, we have $f\text{'}\text{'}(x)=6x$. Thus, for all $x<0$, we also have $f\text{'}\text{'}(x)<0$ and hence, the function $f$ is concave (right curved) on this region. For $x>0$, we have $f\text{'}\text{'}(x)>0$. Hence, for $x>0$, the function $f$ is convex (left curved).