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.
Best Answer
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.