A function f is defined as: f(x) = 2 - |x² + 5x + 6| for x not equal to -2, and f(-2) = a² + 1. For what range of values of a does f(x) attain its maximum value at x = -2?

|a| >= 1
|a| < 1
a > 1
a < 1

Correct answer: |a| >= 1

Solution

Near x = -2: x² + 5x + 6 = (x+2)(x+3) -> 0, so f(x) = 2 - |...| -> 2 for x not equal to -2. For f(-2) = a² + 1 to be the maximum, we need a² + 1 >= 2, i.e., a² >= 1, i.e., |a| >= 1.

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