I'm trying to determine this function concavity:
2=0 is a contradiction, so we have no inflection points on this function, so ¿How could I determine the concavity if I have no inflection points?
This function's graph:
Should I take the "0" as a refered point, then evaluate the f''(x) (for example) with f''(-1) and f''(1) to determine the concavity?
Because on that case, I can effectively determinate the concavity, but is this legal? ¿why I'd take the "0"? On this case, I've take the "0" just because I've seen the graph previously.
$\endgroup$ 61 Answer
$\begingroup$Here $x=0$ is the critical value since $f^{\prime \prime} (0) $ is undefined.
Now use this to divide out your intervals into two intervals.
$(-\infty, 0)$ and $(0, \infty)$.
Pick a test point on each interval and see whether the $f^{\prime \prime}(test value)$ is positive or negative. If it's positive then that mean $f$ is concave up in that interval, and if it's negative then it's concave down.
For example, on the interval, $(-\infty, 0)$ , pick $x=-1$ then $f^{\prime \prime}(-1) = -2$, hence concave down.
$\endgroup$
