Question

Sketch a graph of equation: x^2+2xy+3y^3=4

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Answer 1

[Acknowledgments to Desmos.com for graphing tool.]

Note that the x-intercepts are at x=-2 and x=2, because when y=0, x²=4.

The y-intercept is ∛(4/3), because when x=0, 3y³=4.

The derivative of the equation is 2x+2xy'+2y+9y²y'=0. When y'=0, this becomes 2x+2y=0, so y=-x and:

x²-2x²-3x³=4, 3x³+x²+4=0, which has a zero around x=-1.224, so the maximum of the function is (-1.224,1.224).