Consider an implicitly defined curve in the –plane whose first derivative is finite…

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Consider an implicitly defined curve in the

xy

–plane whose first derivative

\dfrac{dy}{dx}

is finite except possibly at isolated points. Which of the following conditions must hold at a critical point

(x_0, y_0)

of the curve? (a)

\displaystyle \frac{dy}{dx}(x_0,y_0)=0

and

(x_0,y_0)

satisfies the curve’s equation (b) Either

\displaystyle \frac{dy}{dx}(x_0,y_0)=0

\dfrac{dy}{dx}

is undefined at

(x_0,y_0)

, and

(x_0,y_0)

satisfies the curve’s equation (c) Either

\displaystyle \frac{dy}{dx}(x_0,y_0)=0

\dfrac{dy}{dx}

is undefined at

(x_0,y_0)

, regardless of whether

(x_0,y_0)

lies on the curve (d)

\displaystyle \frac{dy}{dx}

is undefined at

(x_0,y_0)

and

(x_0,y_0)

satisfies the curve’s equation