Problem:
Consider functions f that satisfy
∣f(x)−f(y)∣≤21∣x−y∣
for all real numbers x and y. Of all such functions that also satisfy the equation f(300)=f(900), what is the greatest possible value of
f(f(800))−f(f(400))?
Answer Choices:
A. 25
B. 50
C. 100
D. 150
E. 200
Solution:
Note that
f(f(800))−f(f(400))≤∣f(f(800))−f(f(400))∣≤21∣f(800)−f(400)∣=21∣f(800)−f(900)+f(900)−f(300)+f(300)−f(400)∣≤21(∣f(800)−f(900)∣+∣f(900)−f(300)∣+∣f(300)−f(400)∣)≤21(21⋅100+0+21⋅100)=(B)50
This maximum is achieved, for example, by the function
f(x)=−2∣300−∣500−x∣∣+800
Note: A function f for which ∣f(x)−f(y)∣≤c∣x−y∣ for all real numbers x and y is known as a Lipschitz function with constant c.
The problems on this page are the property of the MAA's American Mathematics Competitions