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Let g(x) = x * e^x + 1/(x * e^x) be defined for all x such that x * e^x > 0 (i.e., x > 0). Which of the following statements about g(x) is/are CORRECT?

1. g(x) attains a local maximum at x = x0, where x0 lies in (0, 1)
2. g(x) attains a local minimum at x = -1
3. g(x) attains a local minimum at x = x0, where x0 lies in (1, infinity)
4. g(x) attains a local minimum at x = x0, where x0 lies in (0, 1)

Correct answer: g(x) attains a local minimum at x = x0, where x0 lies in (0, 1)

Solution

Differentiating g(x) = x*e^x + (x*e^x)^(-1) and setting g'(x) = 0 gives (x*e^x)² = 1, so x*e^x = 1 for x > 0. The product x*e^x = 1 has a unique solution x0 in (0, 1) (since at x=0 the product is 0, at x=1 it is e > 1). By the second derivative test or sign change of g', this is a local minimum.

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