Second derivative test operates via first derivative test

Statement

The conclusions for the conclusive cases of the second derivative test follow from the assumptions via the use of the first derivative test.

In particular, this shows that the first derivative test is stronger than the second derivative test. In other words, in any situation where the second derivative test is applicable and conclusive, so is the first derivative test.

Related facts

• Second derivative test is not stronger than first derivative test: This shows that the converse does not hold.

Proof

We provide the proof separately for the two cases where the second derivative test is conclusive, namely, the $f''(c)<0$ case and the $f''(c)>0$ case.