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A function f(x) is defined as f(x) = a*x² + b*x + 2 for x >= 2, and f(x) = 2*a*x² + b*x³ for x < 2. If f is differentiable for all real x, find the value of (|a| + |b|) / 2.

1. 0.6
2. 2.7
3. 1.80
4. 0.75

Correct answer: 0.75

Solution

Setting continuity and differentiability conditions at x=2 gives a system: 4a+2b+2 = 8a+8b and 4a+b = 8a+12b. Solving yields a = -1 and b = 1/2, so (|a|+|b|)/2 = (1+0.5)/2 = 0.75.

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