Consider the curve defined implicitly by

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

Problem

Consider the curve defined implicitly by sin⁡(xy)=x+y\sin(xy)=x+y. The curve intersects the origin. (a) Verify that the origin lies on the curve and calculate dydx\dfrac{dy}{dx} at (0,0)(0,0). Be sure to justify any limits you evaluate. (b) Compute d2ydx2\dfrac{d^{2}y}{dx^{2}} at (0,0)(0,0), showing each differentiation step. (c) Evaluate whether the origin is a point of inflection for the curve. Support your conclusion with appropriate analytical reasoning.

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