Extrema of Implicitly Defined Curves

Question

For a curve defined by an implicit equation, why is a point where $$\frac{dy}{dx}$$ changes sign not necessarily a local extremum?

Accepted Answer

The function may have a vertical tangent line at that point, indicating a change in $$\frac{dy}{dx}$$ without a local extremum.

Answer

The second derivative test always fails for implicit functions at such points.

Answer

$$\frac{dy}{dx}$$ cannot change sign for implicit functions without crossing the x-axis.

Answer

Implicit functions do not have local extrema, only global extrema.

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