Throughout YouTube, I've seen people solve concavity and inflection questions where they find both: concave up and concave down. I was wondering if there were any instances where there was only one of them? Like only a concave up but not a concave down and vice versa?
I have the following function:
$f\left( x\right) =4x^{3}+21x^{2}+36x-20$
According to my work from class, there was no concave up, only a concave down. I'm not sure if this is wrong and I had accidentally missed writing down the part involving the concave up, but if that's not the case, I'm confused as to why there's only a concave down. Could someone explain this to me?
Anyways, this is my work:
$f'\left( x\right) =12x^{2}+42x+36$
$f''(x)=24x+42$
$\begin{aligned}f''(x)=24x+42=0\\ -42=-42\\ \dfrac{24x}{24}=\dfrac{-42}{24}\end{aligned}$
$\begin{aligned}x=-1.75\end{aligned}$
By using the sign chart (via numberline) I've gathered the following:
Concave up: $\left( -1.75, \infty \right)$
Concave down: $\left( -\infty , -1.75\right)$
Is this incorrect?
Your computation looks fine to me.
For the other question, try to look at the log function. It is always concave down.