Let define implicitly as a twice–differentiable function of near a point with .

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Let F(x,y)=0F(x,y)=0 define yy implicitly as a twice–differentiable function of xx near a point P(x0,y0)P(x_{0},y_{0}) with ∂F∂y(x0,y0)≠0\dfrac{\partial F}{\partial y}(x_{0},y_{0})\neq 0. (1) Explain why the existence of dydx\dfrac{dy}{dx} and d2ydx2\dfrac{d^{2}y}{dx^{2}} near PP implies that the graph of yy can be locally approximated by its second–degree Taylor polynomial in xx about x0x_{0}. Your explanation should reference differentiability conditions without reproducing a full proof. (2) Using the local quadratic approximation from part (a), justify how the sign of d2ydx2(x0)\dfrac{d^{2}y}{dx^{2}}(x_{0}) determines whether the graph of yy is concave up or concave down at PP. (3) Describe an additional condition, beyond d2ydx2(x0)=0\dfrac{d^{2}y}{dx^{2}}(x_{0})=0, that must hold to conclude that PP is an inflection point of the curve. Provide a conceptual explanation rather than a computation.

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