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Let y = f(x) be a real-valued function satisfying the differential equation x*(dy/dx) = x² + y - 2, with the initial condition f(1) = 1. Which of the following statements are correct?

1. f(x) has a local minimum at x = 1
2. f(x) has a local maximum at x = 1
3. f(3) = 5
4. f(2) = 3

Correct answer: f(x) has a local minimum at x = 1

Solution

Solving the linear ODE with integrating factor 1/x and applying f(1)=1 gives f(x) = x² - 2x + 2. Then f'(x) = 2x - 2 = 0 at x = 1 with f''(1) = 2 > 0, confirming a minimum. Also f(3) = 9 - 6 + 2 = 5 is correct, while f(2) = 4 - 4 + 2 = 2, not 3.

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