Let $f : x \in \mathbb{R} \rightarrow -x^{4}-2x^{3}+7x$ . Where is it concave?

Question

So I have the following function $f : x \in \mathbb{R} \rightarrow -x^{4}-2x^{3}+7x$ and I have $A$={$\alpha \in \mathbb{R} : $f is concave in $(-\infty,\alpha)$}. I have following alternatives and I have to find which one is correct:

a) $\ supA=1$

b) $\ infA=1$

c) $\ infA=-1$

d) $\ supA= -1$

I thought about it so that I have to find where the second derivative is less than 0, and solving $-12x^{2}-12x <0$ and got $x>-1$ and $x<0$ so I thought that $0$ would be the supremum of A and $-1$ its Infimum.

I would be really thankful for a little bit of help, thanks in advance

Answer 1

It should be $x>0$ or $x<-1$, which gives the answer $d$.

Actually.

$f$ concave on $(-\infty,-1]$ and $f$ concave on $[0,+\infty).$