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Let f : (-π/4, π/4) → R be defined as f(x) = {(1 + |sin x|)^(3a/|sin x|), -π/4 < x < 0 b, x = 0 e^(cot 4x/cot 2x), 0 < x < π/4 If f is continuous at x = 0, then the value of 6a + b^2 is equal to :

1. 1 − e
2. e − 1
3. 1 + e
4. e

Correct answer: e − 1

Solution

The function f is continuous at x = 0 if the limit from both sides equals the value at x = 0. By evaluating the limits as x approaches 0 from the left and right, we find that they both equal e - 1, which must also equal b, leading to the conclusion that 6a + b^2 equals e - 1.

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