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Define p(x) = max{ |x² - 2|x||, |x| } and q(x) = min{ |x² - 2|x||, |x| } for all real x. Which of the following statements is/are true?

1. p(x) is non-differentiable at exactly 5 points in R
2. q(x) is non-differentiable at exactly 7 points in R
3. p(x) is discontinuous and q(x) is continuous in R
4. p(x) is non-differentiable at exactly 6 points in R

Correct answer: p(x) is non-differentiable at exactly 5 points in R

Solution

For x > 0, p(x) = 2x-x² on (0,1), then x on (1,3), then x²-2x for x > 3; corners at x=1 and x=3. By even-function symmetry, also corners at x=-1 and x=-3. At x=0 the left and right derivatives differ (+2 vs -2). So p is non-differentiable at 5 points: {-3,-1,0,1,3}. For q(x), additional non-differentiability appears at x=+-2 because q=|x²-2x| near those points and |x²-2x| has a corner at x=2 where it touches zero, giving 7 total non-diff points.

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