The curve defined implicitly by the equation passes through the point .
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
The curve defined implicitly by the equation x2+xy+y2=7 passes through the point (2,1).
(a) Calculate dxdy at the point (2,1). Show all work.
(b) Determine dx2d2y at the point (2,1). Provide complete algebraic justification.
(c) Based on the value found in part (b), describe the concavity of the curve at (2,1) and explain your reasoning in the context of the second derivative test.