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Show that the curve defined implicitly by the equation xy^3 + x^3y=4 has no horizontal tangent

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you can rewrite your equation as:\[xy(y^2+x^2)=4\]from this we can observe that x or y cannot be zero - agreed?
when and if you come back to this question and if you agree with the statement above, then you should use implicit differentiation to calculate the derivative of the function given to you. you will find that the derivative can never be equal to zero which would imply that the given curve has no horizontal tangents.
OK so I found the derivative

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Do I set it equal to zero ?
yes, tangents are found by setting the derivative to zero and then finding what values satisfy that equation.

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