A twice–differentiable curve C is defined implicitly by an equation G(x,y)=0 and…

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A twice–differentiable curve C is defined implicitly by an equation G(x,y)=0 and passes through the point P(1,2). At P it is known that the first and higher derivatives with respect to x satisfy the following conditions: • dydx∣P=0\displaystyle \frac{dy}{dx}\Big|_{P}=0 • d2ydx2∣P=k\displaystyle \frac{d^{2}y}{dx^{2}}\Big|_{P}=k, where k is a real number that may vary • d3ydx3∣P=6\displaystyle \frac{d^{3}y}{dx^{3}}\Big|_{P}=6 (given only in part (d)) Answer each of the prompts below, providing clear mathematical reasoning in complete sentences and symbolic work. (a) Describe the geometric significance of dydx=0\tfrac{dy}{dx}=0 at P in terms of the tangent line to C. (b) If k=5k=5, determine whether P is a local maximum, local minimum, or neither for y considered as a function of x along C. Justify your conclusion. (c) If k=−4k=-4, classify P in the same manner and justify your reasoning. (d) Suppose k=0k=0. Explain why the usual second–derivative criterion is inconclusive in this case and use the given third derivative to decide whether P is a local extremum. Support your answer with appropriate reasoning.

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