WW: 4
Course: M339D/M389D - Financial Math for Actuaries
Page: 1 of 1
University of Texas at Austin
Warm-up Worksheet # 4
Convexity.
Please, provide a complete solution to the following problem(s):
Problem 4.1. (5 points)
Let xL ≤ x∗ ≤ xR . Express x∗ as a convex combination of xL and xR . More precisely, find λ such that
x∗ = λxL + (1 − λ)xR .
Solution:
λ=
xR − x∗
.
xR − xL
Definition 4.1. A function f : [0, ∞) → R is said to be convex if for every xL < xR and every 0 ≤ λ ≤ 1
we have
f (λxL + (1 − λ)xR ) ≤ λf (xL ) + (1 − λ)f (xR )
Problem 4.2. (5 points)
Provide an example of a convex function f : [0, ∞) → R (by drawing its graph in a coordinate system or
by providing the explicit expression for the function).
Problem 4.3. (10 points)
The increasing function f : [0, ∞) → R satisfies the following inequality for every choice of x1 < x2 < x3
f (x3 ) − f (x2 )
f (x2 ) − f (x1 )
≤
x2 − x1
x3 − x2
Show that the function f is convex.
Solution:
We need to show that for xL < xR and 0 ≤ λ ≤ 1, we have
f (λxL + (1 − λ)xR ) ≤ λf (xL ) + (1 − λ)f (xR ).
Set x1 = xL , x3 = xR and x2 = λx1 + (1 − λ)x3 . Now, it is sufficent to show that
x3 − x2
x2 − x1
f (x2 ) ≤ λf (x1 ) + (1 − λ)f (x3 ) =
f (x1 ) +
f (x3 ).
x3 − x1
x3 − x1
However, this is a straightforward consequence of the given inequality.
Instructor: Milica Čudina