$Functions + mean value theorem$

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$Functions + mean value theorem$

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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 $g(-1)\geq 7$.

Erdos

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$Functions + mean value theorem$

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