An implicitly defined curve yields the derivative

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

An implicitly defined curve yields the derivative dydx=N(x,y)D(x,y)\dfrac{dy}{dx}=\dfrac{N(x,y)}{D(x,y)}, where NN and DD are differentiable at a point (x0,y0)(x_0,y_0). Which of the following conditions is sufficient to guarantee that the curve has a horizontal tangent at (x0,y0)(x_0,y_0)? (a) N(x0,y0)=0N(x_0,y_0)=0 and D(x0,y0)=0D(x_0,y_0)=0 (b) N(x0,y0)=0N(x_0,y_0)=0 and D(x0,y0)≠0D(x_0,y_0)\neq 0 (c) D(x0,y0)=0D(x_0,y_0)=0 and N(x0,y0)≠0N(x_0,y_0)\neq 0 (d) dydx\dfrac{dy}{dx} is undefined at (x0,y0)(x_0,y_0)

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