2020 AMC 12B Problems/Problem 22
What is the maximum value of for real values of
We proceed by using AM-GM. We get . Thus, squaring gives us that . Remembering what we want to find, we divide both sides of the inequality by the positive amount of . We get the maximal values as , and we are done.
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 .
Solution 3 (Calculus Needed)
We want to maximize . We can use the first derivative test. Use quotient rule to get the following: Therefore, we plug this back into the original equation to get
First, substitute so 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 if 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 .
Let the maximum value of the function be . Then we have Solving for , we see We see that Therefore, the answer is .
Upon inspection, the numerator of this expression grows at a relatively faster rate than the denominator, when is close to .
As the numerator is a quadratic in with a negative leading coefficient, its maximum value occurs at or when Therefore,
-Benedict T (countmath1)
~Education, the Study of Everything
Problem starts at 2:10 in this video: https://www.youtube.com/watch?v=5HRSzpdJaX0&t=130s
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