Difference between revisions of "2020 AMC 12B Problems/Problem 22"
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Note that by AM-GM, <cmath>3t(2^t-3t) \leq \frac{2^{2t}}{4}</cmath> | Note that by AM-GM, <cmath>3t(2^t-3t) \leq \frac{2^{2t}}{4}</cmath> | ||
So <cmath>\frac{t(2^t-3t)}{4^t} \leq \frac{1}{12}</cmath> | So <cmath>\frac{t(2^t-3t)}{4^t} \leq \frac{1}{12}</cmath> | ||
− | Equality holds iff <math>2^t =6t</math> which happens for some <math>0<t<1</math> since <math>2^0>6 \times 0 </math> and <math>2^1<6 \times1</math>. So the answer is <math>\boxed{C | + | Equality holds iff <math>2^t =6t</math> which happens for some <math>0<t<1</math> since <math>2^0>6 \times 0 </math> and <math>2^1<6 \times1</math>. So the answer is <math>\boxed{\textbf{(C)}\ \frac{1}{12}}</math> |
==See Also== | ==See Also== |
Revision as of 20:14, 4 March 2020
Contents
[hide]Problem 22
What is the maximum value of for real values of
Solution 1
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 2 (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
~awesome1st
Solution 3
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 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 .
Solution 4 (AM-GM)
Note that by AM-GM, So Equality holds iff which happens for some since and . So the answer is
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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 12 Problems and Solutions |
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