The smiling Buddha

Level pending

For each \( b \in \mathbb{R} \), define

\[f(b) =\max_{x \in \mathbb{R} } ( |b+sin(x)+\frac{2}{3+sin(x)}|)\]

The minimum value of \( f(b) \), is of the form \(\frac{s}{t},\) where \(s\) and \(t\) are coprime integers . Find \(s + t \).

Here \( | \cdot | \) denote modulus function and \( \max( \cdot ) \) denotes the maximum function.

Bonus : A graphical approach to the problem might help you.

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