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**mathproblem** without using a graphing calculator

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How would you find the coordinates of all points(x,y) of the tangent lines of a relation such as (x^2)*y - y^3 = 8 (btw this graph looks like 3 parabolas oppening, one to the right, left and one downwards) where the tangent line is horzontal ?

Would using implicit differentiation work getting y'=(-2xy)/((x^2) -3y^2) then equating it to zero(slope) ?

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Consider x^3 + y^3 - 9xy = 0 ,

how can you isolate the y on one side in the form y = x ?

without using a graphing calculator how would you go about finding the points where the tangent lines to the graph of x^3 + y^3 - 9xy = 0 are horizontal ?

how would you find all the coordinates when y'= (9y -3x^2) / (3y^2 - 9x)= 0 like solving for them algebraicaly