#### Chapter 7 Differential Calculus

Section 7.4 Properties of Functions

# 7.4.3 Second Derivative and Bending Properties (Curvature)

Let us consider a function $f:D\to ℝ$ that is differentiable on the interval $\text{}\right]a;b\left[\text{}\subseteq D$. If its derivative $f\text{'}$ is also differentiable on the interval $\text{}\right]a;b\left[\text{}\subseteq D$, then $f$ is called twice-differentiable. The derivative of the first derivative of $f$ ($\left(f\text{'}\right)\text{'}=f\text{'}\text{'}$) 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 $f\text{'}\text{'}\left(x\right)\ge 0$ for all $x$ between $a$ and $b$, then $f$ is called convex (left curved or concave up) on the interval $\text{}\right]a;b\left[\text{}$.
If $f\text{'}\text{'}\left(x\right)\le 0$ for all $x$ between $a$ and $b$, then $f$ is called concave (right curved or concave down) on the interval $\text{}\right]a;b\left[\text{}$.

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: ${f}^{\left(k\right)}$ 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,
• ${f}^{\left(0\right)}=f$ denotes the function $f$,
• ${f}^{\left(1\right)}=f\text{'}$ denotes the (first) derivative,
• ${f}^{\left(2\right)}=f\text{'}\text{'}$ the second derivative,
• ${f}^{\left(3\right)}$ the third derivative, and
• ${f}^{\left(4\right)}$ the fourth derivative of $f$.
This list can be continued as long as the derivatives of $f$ exist.

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