Functions + mean value theorem 

By mean value theorem, there exists $c \in (-1,2)$ such that 
\[g(2)-g(-1)=g'(c)(2-(-1))\]
\[\Rightarrow 1-g(-1)=3g'(c)  \Rightarrow  g(-1)=1-3g'(3).\]
Since $|g'(c)|\geq 2$,  either $g'(c)\geq 2$ or $g'(c)\leq -2$. 

If $g'(c)\geq 2$, then
\[g(-1)=1-3g'(3) \leq 1-3 \times 2=-5   \Rightarrow g(-1)\leq -5.\]

If $g'(c)\leq -2$, then
\[g(-1)=1-3g'(3) \geq 1-3 \times (-2)=7  \Rightarrow  g(-1)\geq 7.\]

So either $g(-1)\leq -5$ or $g(-1)\geq 7$.