The curve is defined by the equation x^ 3 +y^ 3 -3xy=0

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The curve

C

is defined by the equation

x^{3}+y^{3}-3xy=0.

This curve, often referred to as a folium-type curve, exhibits several interesting stationary behaviors. (a) Find an explicit expression for

\dfrac{dy}{dx}

in terms of

x

and

y

. (b) Determine every critical point of

C

in the finite plane (do not include points at infinity). Clearly indicate whether each critical point corresponds to

\dfrac{dy}{dx}=0

or to

\dfrac{dy}{dx}

being undefined. (c) Compute

\dfrac{d^{2}y}{dx^{2}}

at each finite critical point where the first derivative is defined. Use this information to classify those critical points as local maxima, local minima, or points of inflection along the curve.

Related theory

Curve Analysis
Advanced Differentiation
Implicit and Parametric Differentiation
Derivatives of implicit functions
Continuity and Differentiability

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