2022 AMC 10B Problems/Problem 24
Contents
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 1 (Absolute Values and Inequalities)
We have from which we eliminate answer choices and
Note that Let Together, it follows that We rewrite as ~MRENTHUSIASM
Solution 2 (Lipschitz Condition)
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 .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
~Viliciri (LaTeX edits)
Solution 3 (Slopes)
Divide both sides by to get . This means that when we take any two points on , the absolute value of the slope between the two points is at most .
Let , and since we want to find the maximum value of , we can take the most extreme case and draw a line with slope down from to and a line with slope up from to . Then and , so , and this is attainable because the slope of the line connecting and still has absolute value less than .
Therefore, .
Solution 4 (Translation)
Consider . Then satisfies all the conditions and . We would want and as distant from each other as possible. So assign and , the possible lower and upper bounds respectively. It follows that one can obtain the upper bound for as the answer.
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by OmegaLearn Using Algebraic Manipulation
~ pi_is_3.14
Video Solution by Interstigation
~Interstigation
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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