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For each pair (f, g) where f and g are inverse functions of each other, match Column-I (the function f(x)), Column-II (a value of g'), and Column-III (a value of g'') to identify the only CORRECT combination. Column-I: (I) f(x) = 2x + cos(x); (II) f(x) = x^x on [1/e, infinity); (III) f(x) = arctan(x); (IV) f(x) = e^(x³ + x). Column-II: (i) g'(1) = 1; (ii) g'(4) = 1 / (4(1 + ln 2)); (iii) g'(0) = 1; (iv) g'(1) = 1/2. Column-III: (P) g''(1) = 1/8; (Q) g''(1) = -1; (R) g''(4) = -2; (S) g''(0) = 0.

1. (I) (i) (P)
2. (II) (ii) (R)
3. (III) (i) (S)
4. (IV) (i) (Q)

Correct answer: (II) (ii) (R)

Solution

For f(x) = x^x, f(2) = 4, f'(2) = 4(1 + ln 2), so g'(4) = 1/(4(1+ln2)), matching (ii). Computing g''(4) using the second derivative formula gives -2, matching (R). Hence (II)(ii)(R) is the correct combination.

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