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Chapter 3: Problem 80

Solve the equation algebraically. Round your result to three decimal places. Verify your answer using a graphing utility. $$2 x \ln \left(\frac{1}{x}\right)-x=0$$

Short Answer

Expert verified
The solutions to the equation are \(x = 1\) and \(x = 0.368\), after rounding to three decimal places.

Step by step solution

01

Simplify the Equation

Notice that \(x \ln (1/x) = -x \ln x\) because \(\ln (1/x) = -\ln x\) using logarithm properties, and thus the given equation can be rewritten as: \(2x(-\ln x) - x = -2x \ln x - x = 0\) or \(2x \ln x + x = 0\). So next step is to find \(x\) that satisfies this equation.
02

Solve the Equation

Let's add \(2x \ln x\) to both side to get \(x = -2x \ln x\). Let's note that we can't have \(x = 0\) because \(-2x \ln x\) would be undefined for \(x = 0\). We also note that \(x = 1\) will make the right-hand side zero, so \(x = 1\) is a possible solution. Next, differentiate the right hand side to find the critical point that will give other possible solutions.
03

Differentiate

The derivative of \(-2x \ln x\) using the product rule is \(-2 \ln x - 2\). Setting this to zero and solving for \(x\), we get \(\ln x = -1\). Since the base of natural logarithm is \(e\), we can write this equation in exponential form to get \(x = e^{-1} = 1/e = 0.368\).
04

Analyse and Verification Solutions

By checking the graph or using some reasonable test points, we can determine that \(x = 1\) and \(x = 1/e = 0.368\) are indeed the solutions. Finally, noting that a graphing utility can be used to confirm this.

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