Let $f(x)= \begin{cases}-x^{3}+\frac{\left(b^{3}-b^{2}+b-1\right)}{\left(b^{2}+3 b+2\right)}, & 0 \l

Question

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

Professor PC Thomas Classes · Answer

$b \in(-2,-1) \cup(1, \infty)$\