$g: \mathbb{R} \rightarrow \mathbb{R}$ is 3 times differentiable and $g{'''}(x)>0$ $ \forall x \in \mathbb{R}$ and it has 2 points of extremum $\alpha$ and $\beta$ with $\alpha < \beta$. Tell if $g(\alpha)$ and $g(\beta)$ are maximum or minimum of $g$

My attempt

What I thought was that if the function has extremum for $\alpha$ and $\beta$ that means that $g'(\alpha)=g'(\beta)=0$. By Rolle's theorem that means that there is a $c \in [\alpha, \beta]$, $g''(c)=0$. As $g'''(c)>0$, g has an inflection point in $c$ which means that e can have either $ \alpha $ minima and $\beta$ maxima or $\alpha$ maxima and $\beta$ minima.

Well I'm blocked in this step. But how can I deduce what is the correct answer? Thanks!


The fact that $g'''(x) > 0$ (for all $x$) means that $g''$ is a strictly increasing function. You already found that $g''$ has a zero in $x=c$ with $\alpha < c < \beta$, so $g''(\alpha) < 0$ and $g''(\beta) > 0$, leading to a (local) maximum in $\alpha$ and a (local) minimum in $\beta$ according to the second derivative test.


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