So I was given the following question in my textbook: "Let $f$ be a twice-differentiable function. Selected values of $f'$ and $f''$ are shown in the following table." And then I was told to find which out of the following statements were true:
"$f$ has neither a relative minimum nor a relative maximum at $x=2$"
"$f$ has a relative maximum $x=2$"
"$f$ has a relative maximum $x=8$"
After doing a bit of calculations and applying both the first derivative test, I'm leaning towards the "$f$ has a relative maximum $x=2$" option, since I thought that a sign switch from a $+$ to a $-$ was one indicator of this maximum, along with a possible switch in concavity, which can be seen through the values of the second derivative here. I'm looking for some confirmation here of whether or not this is right, or whether or not I'm completely off. Any feedback would be appreciated!
That's correct. Since $f'(2)=0$ and $f''(2)\not=0$ you can be sure that $f$ has a relative extremum at $2$.
Also $f''(2)<0$ which shows that is a relative maximum by the second derivative test.
However you can't draw conclusions from the change of signs from a $+$ to a $-$, lots of things can happen between $0$ and $1$ which this table does not show.