2022 AMC 10B Problems/Problem 24
Problem
Consider functions that satisfy
for all real numbers
and
. Of all such
functions that also satisfy the equation
, what is the greatest possible value of
Solution
Denote .
Because
,
.
Following from the Lipschitz condition given in this problem, and
and
Thus,
Thus, is maximized at
,
,
, with the maximal value 100.
By symmetry, following from an analogous argument, we can show that is minimized at
,
,
, with the minimal value
.
Following from the Lipschitz condition,
We have already construct instances in which the second inequality above is augmented to an equality.
Now, we construct an instance in which the first inequality above is augmented to an equality.
Consider the following piecewise-linear function:
Therefore, the maximum value of is
\boxed{\textbf{(B) 50}}.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)