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Consider the following functions: (I) f(x) = floor(x) + |x| * sin|x| (II) f(x) = (x² - 1)^(-1) for x² not equal to 1, and f(x) = 0 for x² = 1 (III) f(x) = max(2^x, x*ln2) (IV) f(x) = sin(x*|x|) + sin(pi*(x²-1)/(x²+1)) Where floor(.) denotes the greatest integer function. Match each function with its continuity property (Column II) and differentiability property (Column III): Column II: (i) Continuous for all x in R, (ii) Discontinuous at exactly one point, (iii) Discontinuous at exactly 2 points, (iv) Discontinuous at more than 2 points. Column III: (P) Non-differentiable at exactly one point, (Q) Non-differentiable at exactly two points, (R) Differentiable for all x in R, (S) Non-differentiable at more than two points. Which of the following is the only CORRECT combination?

1. (A) (IV) - (i) - (R)
2. (B) (I) - (i) - (P)
3. (C) (II) - (i) - (R)
4. (D) (IV) - (iv) - (S)

Correct answer: (A) (IV) - (i) - (R)

Solution

f(IV): sin(x|x|) -- here x|x| = x² sgn(x), which equals x² for x>0 and -x² for x<0 and 0 at x=0. The derivative at x=0: lim (x|x|-0)/x = lim |x| = 0, and from the formula d/dx(x²)=2x and d/dx(-x²)=-2x, both give 0 at x=0, so differentiable everywhere. sin(pi*(x²-1)/(x²+1)): x²+1 is never zero, argument is smooth, so this is differentiable everywhere. Hence f(IV) is continuous on R (i) and differentiable on R (R). Answer: (A).

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