Let \( f(x)=\left\{\begin{array}{cc}-x^{3}+\frac{\left(b^{3}-b^{2}+...

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Let \( f(x)=\left\{\begin{array}{cc}-x^{3}+\frac{\left(b^{3}-b^{2}+b-1\right)}{\left(b^{2}+3 b+2\right)}, & 0 \leq x \leq 1 \\ 2 x-3, & 1 \leq x \leq 3\end{array}\right. \)
\( \mathrm{P} \)
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Find all possible real values of \( b \) such that \( f(x) \) has the smallest value at \( x=1 \).
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