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The horizontal line y = α meets the graph of y = g(x) in at least two distinct points. If ∫ from 2 to x of g(t) dt = x²/2 + ∫ from 2 to x of t g(t) dt, then which of the following can be a value of α?
- (-1/2, 1/2)
- [-1/2, 1/2]
- (-1/2, 1/2) \ {0}
- {-1/2, 0, 1/2}
Correct answer: [-1/2, 1/2]
Solution
The correct option is right because the integral equation implies that the function g(x) must oscillate around the horizontal line y = α, allowing for intersections at multiple points. The range [-1/2, 1/2] includes values that can accommodate the behavior of g(x) as indicated by the integral properties, ensuring at least two distinct intersection points.
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