A particle moves along a curve y= 2 x^3-1 3 . The points on the curve at which the y co-ordinate is changing 18 times the x co-ordinate are

Step-by-step solution

The curve is described by y = 2x^3 - 1 3 , with the condition that dy dt = 18 dx dt . Differentiating implicitly with respect to t gives dy dt = 2x^2 dx dt . Equating the two expressions for dy dt yields 18 dx dt = 2x^2 dx dt . Since dx dt 0 , this simplifies to 18 = 2x^2 or x^2 = 9 , giving x = 3 . Substituting into the original equation: at x = 3 , y = 2(27) - 1 3 = 53 3 ; at x = -3 , y = 2(-27) - 1 3 = - 55 3 . The corresponding points are (3, 53 3 ) and (-3, - 55 3 ) , which correspond to option D .