At what points will be tangents to the curve y = 2x³ – 15x² + 36x – 21 be parallel to the x – axis? Also, find the equations of the tangents to the curve at these points.

finding the slope of the tangent by differentiating the curve

According to the question, tangent is parallel to the x – axis , which implies m = 0

x = 3 or x = 2

since this point lies on the curve, we can find y by substituting x

y = 2(3)³ – 15(3)² + 36(3) – 21

y = 6

or

y = 2(2)³ – 15(2)² + 36(2) – 21

y = 7

equation of tangent is given by y – y₁ = m(tangent)(x – x₁)

y – 6 = 0(x – 3)

y = 6

or

y – 7 = 0(x – 2)

y = 7