Question

There is an infamous problem from mathematics that attempts to quantify the number of potential mates one should date before choosing one's "true love." The function $$L(x)=-x \ln x$$ represents the probability of finding the ideal mate after rejecting the first \(x\) proportion of potential mates. For example, if you reject the first \(20 \%=0.20\) of individuals you date, the probability of finding the ideal mate is \(L(0.2) \approx 0.322 .\) So, if you want the probability of finding the ideal mate to be greater than 0.332 and you are only willing to date up to 20 individuals, you should reject the first \(0.2(20)=4\) individuals before attempting to decide on the ideal mate. Presumably, you are using those first 4 individuals to help you decide which traits you value in a mate. (a) Determine and interpret \(L(0.1)\). (b) Determine and interpret \(L(0.6)\) (c) What is the domain of \(L ?\) (d) Graph \(L=L(x)\) over the domain. (e) Judging on the basis of the approach suggested by the model, what is the value of \(x\) that maximizes \(L\) ? What is the highest probability of finding the ideal mate?

Short answer

L(0.1) = 0.2302, L(0.6) = 0.306, domain is (0, 1], maximum probability at x = 0.3679, with value 0.3679.

Step-by-step solution

Determining and Interpreting L(0.1)

To find the value of \(L(0.1)\), substitute \(x = 0.1\) into the function \(L(x) = -x \ln x\). Compute \(L(0.1) = -0.1 \ln(0.1)\). Use the natural logarithm properties: \(\ln(0.1) = \ln(1/10) = -\ln(10)\). Therefore, \(L(0.1) = -0.1 (-\ln(10)) = 0.1 \ln(10)\). Since \(\ln(10) \approx 2.302\), \(L(0.1) \approx 0.1 \times 2.302 = 0.2302\). Interpret this result: The probability of finding the ideal mate after rejecting the first 10% of individuals is approximately 0.2302 or 23.02%.

Determining and Interpreting L(0.6)

To find the value of \(L(0.6)\), substitute \(x = 0.6\) into the function \(L(x) = -x \ln x\). Compute \(L(0.6) = -0.6 \ln(0.6)\). Since \(\ln(0.6) \approx -0.51\), \(L(0.6) = -0.6 \times -0.51 = 0.306\). Interpret this result: The probability of finding the ideal mate after rejecting the first 60% of individuals is approximately 0.306 or 30.6%.

Determining the Domain of L(x)

The domain of the function \(L(x) = -x \ln x\) consists of the values of \(x\) where the function is defined. Since the natural logarithm \(\ln(x)\) is only defined for \(x > 0\), the domain of \(L(x)\) is \((0, 1]\).

Graphing L(x) over the domain

Plotting the function \(L(x) = -x \ln x\) over the domain \(x \in (0, 1]\): The graph starts at (0,0), increases, reaches a maximum, and then decreases towards zero as \(x\) approaches 1. The function is concave down.

Finding the Maximum Value of L(x)

To find the value of \(x\) that maximizes \(L(x)\), take the derivative of \(L(x) = -x \ln x\) with respect to \(x\) and set it to zero. The derivative is \(L'(x) = -\ln x - 1\). Set \(L'(x) = 0\): \(-\ln(x) - 1 = 0\), solving yields \(\ln(x) = -1\), so \(x = e^{-1} = \frac{1}{e} \approx 0.3679\). Compute \(L(\frac{1}{e}) = -\frac{1}{e} \ln(\frac{1}{e}) = \frac{1}{e}\). Therefore, the maximum probability of finding the ideal mate is \(\frac{1}{e} \approx 0.3679\) or 36.79%.

Key concepts

Logarithmic Functions

Logarithmic functions are a key component in mathematics and appear in various applications. The function used in the highlighted problem, \(L(x) = -x \, \ln x\), relies heavily on the natural logarithm (\(\ln\)). Logarithms transform multiplicative relationships into additive ones, which often simplifies complex calculations. For example, \(\ln(10) \approx 2.302\), which means that the number 10 is equivalent to \(e^{2.302}\) in exponential terms.
Understanding logarithms is essential for interpreting the behavior of such functions. They are the inverse of exponential functions, so \(y = \ln x\) implies \(e^y = x\). This relation is handy for solving equations involving exponential growth or decay.
Moreover, the natural logarithm \(\ln x\) is only defined for values where \(x > 0\). Hence, when working with logarithmic functions, it is critical to ensure that all inputs fall within this range to avoid undefined results.

Domain and Range

The domain of a function consists of all the possible input values while the range consists of all possible output values.
For the function \(L(x) = -x \, \ln x\), the domain is determined mainly by the argument of the logarithm. Since \(\ln x\) is only defined for \(x > 0\), the domain of \(L(x)\) is \((0, 1]\). This means you can only input values between 0 (not including 0) and 1 (including 1).
The range of \(L(x)\) focuses on the output values the function can take. Given the nature of \(-x \ln x\), the values will start at 0 at both ends of the domain, reaching a maximum in between. This means the range is \([0, \frac{1}{e}]\), where \(\frac{1}{e}\) is the maximum value of the function and approximately equals 0.3679.

Derivatives

Derivatives are central to calculus and are used to determine the rate at which a function changes. To find the maximum or minimum values of a function, you typically take its derivative and find where this derivative equals zero.
For the function \(L(x) = -x \ln x\), we calculated its derivative to find the maximum value. The derivative is found using the product rule and is \(L'(x) = - \ln x - 1\). Setting \(L'(x) = 0\), we solve for \(x\), resulting in \(x = \frac{1}{e}\). This critical point represents the value of \(x\) that maximizes the function, which is crucial for solving optimal stopping problems.
Interpreting derivatives helps identify points where the function's value changes direction, such as turning from increasing to decreasing, representing a maximum.

Probability Theory

Probability theory helps quantify uncertainty and formulate strategies under uncertain conditions. In the given problem, the function \(L(x) = -x \, \ln x\) represents the probability of finding an ideal mate after rejecting a portion of potential mates.
Interpreting this probability involves understanding that the function value, such as \(L(0.1) = 0.2302\), indicates the likelihood (23.02%) of success given specific conditions (rejecting the first 10%).
Probability theory is crucial in decision-making processes, guiding individuals on the chances of successful outcomes under different scenarios. In this context, it helps determine the optimal strategy for maximizing success in mate selection.

Graphing Functions

Graphing functions provides visual insights that are not always evident from equations alone. By graphing \(L(x) = -x \, \ln x\), we can observe how the function behaves across its domain.
The graph starts at (0,0), increases to a peak, and then decreases towards zero as \(x\) approaches 1. This shape is concave down, indicating the function's maximum point in the middle. By analyzing the graph, one can better understand the relationship between the inputs (rejected proportions) and the resulting probabilities.
Graphical representations are powerful tools for visual learning, aiding in understanding and interpreting the behavior of mathematical functions quickly and effectively.