The differential equation defines a slope field

MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations

Problem

The differential equation dydx=y2−x\dfrac{dy}{dx}=y^{2}-x defines a slope field. For the solution curve that passes through (1,1)(1,1), which statement best describes the concavity of the solution at that point? (a) Concave upward because d2ydx2=1>0\dfrac{d^{2}y}{dx^{2}}=1>0 at (1,1)(1,1).},{ (b) Point of inflection because d2ydx2=0\dfrac{d^{2}y}{dx^{2}}=0 at (1,1)(1,1).},{ (c) Concave downward because d2ydx2=−1<0\dfrac{d^{2}y}{dx^{2}}=-1<0 at (1,1)(1,1).},{ (d) Cannot be determined without explicitly solving for y(x)y(x).}],

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