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Given f(x) = ((x - 1)² * (x - 2)³) / e^x, which of the following correctly describes the behaviour of f(x)?

1. local maximum at x = 1
2. point of inflection at x = 1
3. local minima at x = 2
4. point of inflection at x = 2

Correct answer: local maximum at x = 1

Solution

f'(x) = [(x-1)(x-2)²(5-x)] / e^x. At x=1, f' changes sign from negative to positive giving a local minimum (not maximum); but since the factor (x-1)² makes x=1 a double root with no sign change in f', x=1 is actually a point of inflection. At x=2, f' does not change sign (even power factor), so x=2 is also a point of inflection.

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