First Derivative Test vs Second Derivative Test for Local Extrema
Key Questions

Since you are asking for the difference, I assume that you are familiar with how each test works. The biggest difference is that the first derivative test always determines whether a function has a local maximum, a local minimum, or neither; however, the second derivative test fails to yield a conclusion when
#y''# is zero at a critical value. If that is the case, you will have to apply the first derivative test to draw a conclusion.
Questions
Graphing with the Second Derivative

Relationship between First and Second Derivatives of a Function

Analyzing Concavity of a Function

Notation for the Second Derivative

Determining Points of Inflection for a Function

First Derivative Test vs Second Derivative Test for Local Extrema

The special case of x⁴

Critical Points of Inflection

Application of the Second Derivative (Acceleration)

Examples of Curve Sketching