Find the points on the curve y = (cosx – 1) in [0, ], where the tangent is parallel to x-axis.

Given: Equation of curve, y = cos x – 1

Firstly, we differentiate the above equation with respect to x, we get

Given tangent to the curve is parallel to the x – axis

This means, Slope of tangent = Slope of x – axis

⇒ - sin x = 0

⇒ sin x = 0

⇒ x = sin^-1(0)

⇒ x = π ∈ (0, 2π)

Put x = π in y = cos x – 1, we have

y = cos π – 1 = -1 – 1 = -2 [∵ cos π = -1]

Hence, the tangent to the curve is parallel to the x –axis at

(π, -2)