2. The Second Derivative and the Second Derivative Test

The second derivative is crucial for differentiating between
a maximum, minimum, or saddle point at a critical point.
This is known as the Second Derivative Test.

* f''(x) > 0 (Convexity / Concave Up): This indicates the
function is locally convex (shaped like a bowl opening upward).
At a critical point where f'(x) = 0 and f''(x) > 0, the point
is a local minimum.

* f''(x) < 0 (Concavity / Concave Down): This indicates the
function is locally concave (shaped like a dome opening downward).
At a critical point where f'(x) = 0 and f''(x) < 0, the point
is a local maximum.

* f''(x) = 0: The test is inconclusive, and the point might be
an inflection point or a saddle point.