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Given C(1) lt C(2) lt C(3) lt C(4) lt C(...

Given `C_(1) lt C_(2) lt C_(3) lt C_(4) lt C_(5)` and the function `y=f(x)` is twice differentiable .
`f'(x) gt 0 " for " x in (C_(2),C_(4)), f'(C_(2))=f'(C_(4))=0` and `f'(x) lt 0` for all the remaining values. Also `f''(C_(1))=f''(C_(3))=f''(C_(5))=0` and `f''(x) gt 0 ` for `x in (C_(1),C_(3)) cup (C_(5), oo)` and `f''(x) lt 0` for remaining values. Answer the following:
(i) What is the minimum number of asymptotes parallel to the x-axis for `y=f(x)`?
(ii) What is the maximum number of asymptotes parallel to the x-axis of `y=f(x)` ?
(iii) If the range of `y=f(x)` is `[a, b], a , b in R`, then what is the minimum number of asymptotes parallel to the x-axis of `y=f(x)` ?
(iv) If the range of `y=f(x)` is non-finite interval, then what is the maximum number of asymptotes parallel to the x-axis ?

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To solve the given problem, we will analyze the behavior of the function \( y = f(x) \) based on the information provided about its first and second derivatives. ### Step-by-step Solution: 1. **Understanding the Given Information**: - We have critical points \( C_1, C_2, C_3, C_4, C_5 \) such that \( C_1 < C_2 < C_3 < C_4 < C_5 \). - The first derivative \( f'(x) \) is positive in the interval \( (C_2, C_4) \), indicating that the function is increasing in this interval. - At \( C_2 \) and \( C_4 \), \( f'(C_2) = 0 \) and \( f'(C_4) = 0 \), indicating local extrema (minimum at \( C_2 \) and maximum at \( C_4 \)). ...
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