The curve defined implicitly by contains branches on which can be expressed as a…

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

Problem

The curve defined implicitly by

x^{2}+xy+y^{2}=7

contains branches on which

y

can be expressed as a differentiable function of

x

. Focus on the branch for which

y>0

when

0<x<2

. (a) Derive an expression for

\dfrac{dy}{dx}

in terms of

x

and

y

. (b) Using your result from part (a), find an explicit formula for

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

in terms of

x

and

y

. (c) For

0<x<2

with

y>0

, determine the sign of

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

. Support your answer algebraically. (d) State the resulting concavity of the branch described above on the interval

0<x<2

, and explain how your conclusion follows from part (c).

Related theory

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

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