Mathematics

Mathematics

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.
Intuitive explanation

To find concavity, you need the second derivative d2ydx2\frac{d^2y}{dx^2} at that point. Use implicit differentiation twice on the curve equation: first to get dydx\frac{dy}{dx}, then differentiate again to find d2ydx2\frac{d^2y}{dx^2}. The sign of the second derivative tells you whether the curve bends upward (concave up) or downward (concave down) at (1,2)(1,2).

Intuitive explanation

To find concavity, you need the second derivative d2ydx2\frac{d^2y}{dx^2} at that point. Use implicit differentiation twice on the curve equation: first to get dydx\frac{dy}{dx}, then differentiate again to find d2ydx2\frac{d^2y}{dx^2}. The sign of the second derivative tells you whether the curve bends upward (concave up) or downward (concave down) at (1,2)(1,2).

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