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This problem trains for: AMC-10, AMC-12, AIME.
For what values of the real parameter b does the equation:
have at least one root in the interval (-1000, -995)?
Notice that the translation:
will reduce the problem to a similar one with smaller constants, a possible simplification.
After making this change of variable:
The function associated with this equation is the cubic polynomial:
and the interval in which we must place the root is:
For the function to have at least one root in the given interval it has to satisfy the mean-value theorem:
Solve the inequality: