# 2020 AMC 12B Problems/Problem 22

## Problem 22

What is the maximum value of for real values of

## Solution1

Set . Then the expression in the problem can be written as It is easy to see that is attained for some value of between and , thus the maximal value of is .

## Solution2

First, substitute so that

Notice that

When seen as a function, is a synthesis function that has as its inner function.

If we substitute , the given function becomes a quadratic function that has a maximum value of when .

Now we need to check that can have the value of in the range of real numbers.

In the range of (positive) real numbers, function is a continuous function whose value gets infinitely smaller as gets closer to 0 (as also diverges toward negative infinity in the same condition). When , , which is larger than .

Therefore, we can assume that equals to when is somewhere between 1 and 2 (at least), which means that the maximum value of is .