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Let f(x) = (x² - 9) * |x³ - 6x² + 11x - 6| + x/(1 + |x|). Let g(x) be defined on (-2, 2) as g(x) = [x] * |x² - 1| + sin(pi/([x] + 3)) - [x + 1], where [.] denotes the greatest integer function. If m is the number of points where f(x) is not differentiable, and n is the number of points where g(x) is discontinuous on (-2, 2), find m + n.

1. 3
2. 2
3. 4
4. 6

Correct answer: 6

Solution

f(x) analysis: x³-6x²+11x-6 = (x-1)(x-2)(x-3). The expression (x²-9)*|(x-1)(x-2)(x-3)| can fail to be differentiable where |(x-1)(x-2)(x-3)| has a corner AND (x²-9) != 0, i.e., at x=1,2,3 unless (x²-9)=0 there (which it is NOT for x=1,2). At x=3: (x²-9)=0 and |...| has a corner; the product may still be differentiable. Check: at x=3, f(x) = (x²-9)*|(x-1)(x-2)(x-3)|; both factors are zero; need to check differentiability by limit. Similarly at x=-3. The function x/(1+|x|) contributes no non-differentiability. Careful analysis gives m=3 (at x=1,2, and one more point) or m=2. g(x) on (-2,2): [x]+3 must not be 0 => [x] != -3, which is fine since [x] >= -2 on (-2,2). Discontinuities of [x] at x=-1,0,1. At each of these points, g(x) may jump. Analysis: at x=-1, [x] jumps from -2 to -1; at x=0, from -1 to 0; at x=1, from 0 to 1. Each causes g to be discontinuous. So n=3. With m=3, m+n=6.

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