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→ℝ that is differentiable on the interval ]a;b[ ⊆D. If its derivative f' is also differentiable on the interval ]a;b[ ⊆D, then f is called twice-differentiable. The derivative of the first derivative of f ( (f')'=f'') 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''(x)≥0 for all x between a and b, then f is called convex (left curved or concave up) on the interval ]a;b[ .
If f''(x)≤0 for all x between a and b, then f is called concave (right curved or concave down) on the interval ]a;b[ .

Thus, it is sufficient to determine the sign of the second derivative f'' 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 kth 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(0) =f denotes the function f,
  • f(1) =f' denotes the (first) derivative,
  • f(2) =f'' the second derivative,
  • f(3) the third derivative, and
  • f(4) 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→ℝ,x→ x3 is at least twice-differentiable. Since f'(x)=3 x2 ≥0 for all x∈ℝ, the function f is monotonically increasing on its entire domain. Moreover, we have f''(x)=6x. Thus, for all x<0, we also have f''(x)<0 and hence, the function f is concave (right curved) on this region. For x>0, we have f''(x)>0. Hence, for x>0, the function f is convex (left curved).

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