Let be the set of points satisfying the implicit relation e^ x +e^ y =x^ 2 y^ 2

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

Problem

Let

S

be the set of points satisfying the implicit relation

e^{x}+e^{y}=x^{2}y^{2}.

The curve encloses a bounded region in the first quadrant. (a) Derive

\dfrac{dy}{dx}

in terms of

x

and

y

. (b) Show that every critical point of

S

in the first quadrant lies on the line

y=x

. Justify each step of your reasoning. (c) Prove that

S

possesses exactly one critical point in the first quadrant and determine its coordinates to three decimal places. (d) Evaluate

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

at the critical point found in part (c) and discuss what this indicates about the nature of the curve near that point.