At the point on the curve

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

Problem

At the point (1,2)(1,2) on the curve x2+xy+y2=7x^{2}+xy+y^{2}=7, what can be concluded about the concavity of the curve? (a) It is concave up because d2ydx2=42125>0\displaystyle \frac{d^{2}y}{dx^{2}}=\tfrac{42}{125}>0 at that point. (b) It is concave down because d2ydx2=−42125<0\displaystyle \frac{d^{2}y}{dx^{2}}=-\tfrac{42}{125}<0 at that point. (c) The concavity is undefined because the second derivative does not exist at that point. (d) It has an inflection point because d2ydx2=0\displaystyle \frac{d^{2}y}{dx^{2}}=0 at that point.

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