762426…: “Near to a light mountain”

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Consider the function $F(x,y) = x^2y-xy^2+4$ . A representation of several curves are given below. Select all that are a part of a level curve of $F(x,y)$ .

$y=x$ $y=2x$ $x^2y-xy^2=7$ $\vec {r}(t) =\vector {0,5t^2+1}$ $\vec {r}(t) =\vector {4t,4t}$ $\vec {r}(t)=\vector {7,t}$

Evaluate $F(x,y)$ along each given curve. For example, along $y=2x$ , $F(x,y)=F(x,2x)$ (since $y=2x$ ) so

$F(x,2x) = x^2(2x)-x(2x)^2+4 = \answer {-2}x^3+4.$ Since $F(x,y)$ is along the curve $y=2x$ , this curve part of a level curve of $F(x,y)$ .