Continuity and DifferentiabilityHard
Question
If f(x) = (ax + b) sin x + (cx + d) cos x, then the values of a, b, c and d such that f′(x) = x cos x for all x are
Options
A.a = d = 1
B.b = 0
C.c = 0
D.b = c
Solution
f′(x) = a sin x + (ax + b) cos x + c cos x - (cx + d) sin x = x cos x
⇒ (a - cx - d) sin x + (ax + b + c) cos x = x cos x
⇒ a ≠ 0 and a = 1 and b + c = 0 and a - cx - d = 0
⇒ a = 1, b = - c and cx = a - d
⇒ c = 0
⇒ b = c = 0 and a = d = 1
⇒ (a - cx - d) sin x + (ax + b + c) cos x = x cos x
⇒ a ≠ 0 and a = 1 and b + c = 0 and a - cx - d = 0
⇒ a = 1, b = - c and cx = a - d
⇒ c = 0
⇒ b = c = 0 and a = d = 1
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