Consider the implicit curve defined by 3x^ 2 +2xy+y^ 2 =10

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Consider the implicit curve defined by

3x^{2}+2xy+y^{2}=10.

Throughout this question, restrict attention to points on the curve for which

y

can be regarded locally as a differentiable function of

x

. (1) Derive an expression for

\dfrac{dy}{dx}

in terms of

x

and

y

. Simplify your result. (2) Identify all points on the curve at which the

y

–coordinate could attain a relative extremum with respect to

x

. Include both categories: points where

\dfrac{dy}{dx}=0

and points where

\dfrac{dy}{dx}

is undefined but the curve is smooth. (3) For each point you found in part (b), determine whether it corresponds to a relative minimum, a relative maximum, or neither for the

y

–coordinate as

x

varies. Justify your conclusions by examining the sign of

\dfrac{dy}{dx}

on intervals immediately to the left and right of each

x

–value. (4) Describe in complete sentences how the behavior of the curve near each extremum you classified in part (c) is consistent with your derivative sign analysis. Then, on axes of your own, sketch a small neighborhood of the curve around each extremum and clearly label the points you identified.

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