The number of points on the curve y = 54 x 5 - 135 x 4 - 70 x 3 + 180 x 2 + 210 x at which the normal lines are parallel to x + 90 y + 2 = 0 is:

Step-by-step solution

Given curve is y = 54 x 5 - 135 x 4 - 70 x 3 + 180 x 2 + 210 x Normal lines on this curve, are parallel to x + 90 y + 2 = 0 , hence slope of normal is m = - 1 90 , so slope of tangent to the given curve is - 1 m = 90 . Slope of tangent on the given curve is d y d x = 270 x 4 - 540 x 3 - 210 x 2 + 360 x + 210 Here, d y d x = 90 ⇒ 270 x 4 - 540 x 3 - 210 x 2 + 360 x + 210 = 90 ⇒ 270 x 4 - 540 x 3 - 210 x 2 + 360 x + 120 = 0 ⇒ 27 x 4 - 54 x 3 - 21 x 2 + 36 x + 12 = 0 ⇒ 9 x 2 - 18 x 3 - 7 x 2 +