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How many times does the curve y = x⁵ - 20x³ + 50x + 2 cross the x-axis?

1. 1
2. 2
3. 3
4. 4

Correct answer: 3

Solution

The derivative f'(x) = 5x⁴ - 60x² + 50 has four real roots (discriminant of x⁴-12x²+10 is positive), giving four critical points and hence five monotone intervals. Evaluating signs at the critical points confirms the curve crosses the x-axis exactly three times.

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