2nd Derivative Test Option 1: Intervals of Concavity and Inflection Points

Meaning: 2 ^nd Derivative, $f\prime \prime \left(x\right)$.

The 2 ^nd Derivative, $f\prime \prime \left(x\right)$, provides you information about the concavity of your original equation, $f\left(x\right)$.

Just like with the 1 ^st Derivative, it all boils down to positive, negative, or zero.

Definition: Inflection Point

All inflection points must satisfy two requirements:

1)      $f\prime \prime \left(x\right)=0$ or $f\prime \prime \left(x\right)= \mathit{Does\; Not\; Exist }\left(\mathit{DNE}\right)$

AND

2)      The function, $f\left(x\right)$, must change concavity at that point.

The graph must change from concave up to concave down or concave down to concave up.

Note: It is absolutely possible for you to find the $f\prime \prime \left(x\right)=0$ or $f\prime \prime \left(x\right)= \mathit{Does\; Not\; Exist }\left(\mathit{DNE}\right)$, but the graph does not change concavity at that point. This is why you always need to do a 2 ^nd Derivative Test to determine if an x-value is actually an inflection point.

An example of this would be $f\left(x\right)=x^4$. At x = 0 the 2 ^nd Derivative would equal zero, $f\prime \prime \left(0\right)=0$, but you can see that the graph does not change concavity at x = 0 . The graph is always concave up.