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This problem trains for: AMC-10, AMC-12, AIME.

For what values of the real parameter b does the equation:

equation

have at least one root in the interval (-1000, -995)?

Notice that the translation:

equation

will reduce the problem to a similar one with smaller constants, a possible simplification.

After making this change of variable:

equation

equation

The function associated with this equation is the cubic polynomial:

equation

and the interval in which we must place the root is:

equation

For the function to have at least one root in the given interval it has to satisfy the mean-value theorem:

equation

Solve the inequality:

equation

equation