EXAMPLE 2 (a) Find y' if x3 + y3 = 18xy. (b) Find the tangent to the folium of Descartes x3 + y3 = 18xy at the point (9, 9). (c) At what point in the first quadrant is the tangent line horizontal?

(a) Via implicit differentiation, we get

Solve for :

(b) Find the slope of the tangent line at (9, 9) by plugging in x = y = 9 into the equation above:

Use the point-slope formula to find the equation of the line:

(c) The tangent line is horizontal when its slope is 0, so solve :

Now substitute y in the equation for the folium to solve for x :

x = 0 corresponds to y = 0 (plug x = 0 into the folium equation to see why), i.e. the origin. If you don't consider the origin to belong to the first quadrant, then we only keep

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where are the graphs?

step-by-step explanation: