The curve C has the equation: 16y^3 +9x^2y-54x=0, find the x coordinates of the points on C where dy/dx = 0
C: 16y3 + 9x2y -54x = 0 d/dx(16y3) + d/dx(9x2y) + d/dx(-54x) = d/dx(0).............=>48y2.dy/dx + 9x2.dy/dx + y.18x - 54 = 0 dy/dx(48y2+9x2) = 54 - 18xy...........................................=> dy/dx = (54-18xy)/(48y2+9x2) = 0 # dy/dx=0 at turning points therefore: 54-18xy = 0 => y = 3/x # substitute y=3/x into C:=>16.(3/x)3 + 9x2(3/x) -54x = 0 = 432/x3 + 27x -54x = 0 = 432/x3 -27x = 0= 432/x3 = 27x = 432 = 27x4 = 6 = x4 ................................................=> x = 2, -2
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