An implicitly defined curve passes through the point

MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation

Problem

An implicitly defined curve passes through the point (1,2)(1,2). At that point the first derivative satisfies dydx=0\dfrac{dy}{dx}=0 and the second derivative satisfies d2ydx2<0\dfrac{d^{2}y}{dx^{2}}<0. Which of the following statements is most consistent with this information? (a) The curve has a vertical tangent and is concave down at (1,2)(1,2). (b) The curve has a horizontal tangent and is concave up at (1,2)(1,2). (c) The curve has a horizontal tangent and is concave down at (1,2)(1,2). (d) The curve has slope −5-5 and is concave down at (1,2)(1,2).

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