For each constant

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

Problem

For each constant

k\in\mathbb{R}

, consider the family of curves defined implicitly by

x y=\ln(y)+k,

where

y>0

so that

\ln(y)

is defined. Assume

y

can be locally expressed as a differentiable function of

x

. (a) Derive a fully simplified formula for

\dfrac{dy}{dx}

in terms of

x

and

y

. (b) Identify all points

(x,y)

on any member of the family where the curve possesses a vertical tangent line. Express your answer in terms of

k

and justify that the tangent is indeed vertical. (c) For the specific curve corresponding to

k=-1

, determine whether any vertical tangent lines exist. Provide a complete justification rooted in the derivative you found in part (a). (d) For general

k

, assess whether there exists a value of

k

such that the corresponding curve has both at least one horizontal tangent and at least one vertical tangent (not necessarily at the same point). Give a thorough argument that cites necessary conditions on

\dfrac{dy}{dx}

and relates them to the implicit equation.

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