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Let f(x) = (sin x + cos x - sqrt(2)) / (sin x - cos x) for x in [0, pi] excluding x = pi/4. The value of f''(7*pi/12) * f'''(7*pi/12) is equal to

1. (A) -2/3
2. (B) 2/9
3. (C) -1/(3*sqrt(3))
4. (D) 4*sqrt(3)/27

Correct answer: (D) 4*sqrt(3)/27

Solution

By substituting u = x - pi/4, the function simplifies to f(x) = -tan(x/2 - pi/8). Successive differentiation using the chain rule, evaluated at g = pi/6 where tan(pi/6) = 1/sqrt(3) and sec²(pi/6) = 4/3, gives f'' = -2/(3*sqrt(3)) and f''' = -2/3, whose product is 4*sqrt(3)/27.

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