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Let f(x) = x¹¹ + sin³(35x) + 111*x. Find the value of f^(-1)(sin(pi/5)) + f^(-1)(sin(6*pi/5)) + f^(-1)(sin(pi/7)) + f^(-1)(sin(8*pi/7)).

1. 0
2. 1
3. -1
4. pi

Correct answer: 0

Solution

f(x) = x¹¹ + sin³(35x) + 111x is an odd function since all terms are odd functions of x. It is strictly increasing (derivative = 11x¹⁰ + 105*sin²(35x)*cos(35x) + 111 > 0 for large enough constant term 111). So f^(-1) exists and is also odd: f^(-1)(-y) = -f^(-1)(y). Now: sin(6*pi/5) = sin(pi + pi/5) = -sin(pi/5). So f^(-1)(sin(6*pi/5)) = f^(-1)(-sin(pi/5)) = -f^(-1)(sin(pi/5)). Similarly sin(8*pi/7) = sin(pi + pi/7) = -sin(pi/7). So f^(-1)(sin(8*pi/7)) = -f^(-1)(sin(pi/7)). Sum = f^(-1)(sin(pi/5)) - f^(-1)(sin(pi/5)) + f^(-1)(sin(pi/7)) - f^(-1)(sin(pi/7)) = 0.

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