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Difference between revisions of "2020 AMC 12B Problems/Problem 22"

(Created page with "==Problem 22== What is the maximum value of <math>\frac{(2^t-3t)t}{4^t}</math> for real values of <math>t?</math> <math>\textbf{(A)}\ \frac{1}{16} \qquad\textbf{(B)}\ \frac{...")
 
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<math>\textbf{(A)}\ \frac{1}{16} \qquad\textbf{(B)}\ \frac{1}{15} \qquad\textbf{(C)}\ \frac{1}{12} \qquad\textbf{(D)}\ \frac{1}{10} \qquad\textbf{(E)}\ \frac{1}{9}</math>
 
<math>\textbf{(A)}\ \frac{1}{16} \qquad\textbf{(B)}\ \frac{1}{15} \qquad\textbf{(C)}\ \frac{1}{12} \qquad\textbf{(D)}\ \frac{1}{10} \qquad\textbf{(E)}\ \frac{1}{9}</math>
  
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==Solution==
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==Solution1==
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Set <math> u = t2^{-t}</math>. Then the expression in the problem can be written as <cmath>t2^{-t} - 3t4^{-t} = u - 3u^2 = \frac{1}{12}- 3 (\frac{1}{6} - u)^2.</cmath> The maximal value is thus <math>\frac{1}{12}</math>.
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==Solution2==
  
 
First, substitute <math>2^t = x (log_2{x} = t)</math> so that  
 
First, substitute <math>2^t = x (log_2{x} = t)</math> so that  

Revision as of 23:57, 7 February 2020

Problem 22

What is the maximum value of $\frac{(2^t-3t)t}{4^t}$ for real values of $t?$

$\textbf{(A)}\ \frac{1}{16} \qquad\textbf{(B)}\ \frac{1}{15} \qquad\textbf{(C)}\ \frac{1}{12} \qquad\textbf{(D)}\ \frac{1}{10} \qquad\textbf{(E)}\ \frac{1}{9}$

Solution1

Set $u = t2^{-t}$. Then the expression in the problem can be written as \[t2^{-t} - 3t4^{-t} = u - 3u^2 = \frac{1}{12}- 3 (\frac{1}{6} - u)^2.\] The maximal value is thus $\frac{1}{12}$.

Solution2

First, substitute $2^t = x (log_2{x} = t)$ so that \[\frac{(2^t-3t)t}{4^t} = \frac{xlog_2{x}-3(log_2{x})^2}{x^2}\]

Notice that \[\frac{xlog_2{x}-3(log_2{x})^2}{x^2} = \frac{log_2{x}}{x}-3(\frac{log_2{x}}{x})^2.\]

When seen as a function, $\frac{log_2{x}}{x}-3(\frac{log_2{x}}{x})^2$ is a synthesis function that has $\frac{log_2{x}}{x}$ as its inner function.

If we substitute $\frac{log_2{x}}{x} = p$, the given function becomes a quadratic function that has a maximum value of $\frac{1}{12}$ when $p = \frac{1}{6}$.


Now we need to check that $\frac{log_2{x}}{x}$ can have the value of $\frac{1}{6}$ in the range of real numbers.

In the range of (positive) real numbers, function $\frac{log_2{x}}{x}$ is a continuous function whose value gets infinitely smaller as $x$ gets closer to 0 (as $log_2{x}$ also diverges toward negative infinity in the same condition). When $x = 2$, $\frac{log_2{x}}{x} = \frac{1}{2}$, which is larger than $\frac{1}{6}$.

Therefore, we can assume that $\frac{log_2{x}}{x}$ equals to $\frac{1}{6}$ when $x$ is somewhere between 1 and 2 (at least), which means that the maximum value of $\frac{(2^t-3t)t}{4^t}$ is $\textbf{(C)}\ \frac{1}{12}$.

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