A one-parameter family of plane curves is given by

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

y^{2}=x^{3}+k x^{2}

, where

k

is a real constant. (a) Compute

\dfrac{dy}{dx}

in terms of

x

y

, and

k

. (b) Determine the value(s) of

k

for which the curve has a horizontal tangent line at the origin

(0,0)

. Justify your conclusion by verifying that

(0,0)

lies on the curve for those

k

and satisfies the required tangency condition. (c) For a value of

k

that admits a horizontal tangent at the origin, analyze the local behavior of the curve near the origin and describe whether the origin represents a point of inflection, a cusp, or neither.

Related theory

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

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