find the point x

Question

find the point x, 0 x 1, on the graph of y = sin²(3x) where the tangent line is horizontal.

x =



Answer 1

y = sin²(3x)

Horizontal tangent when y' = 0.

y' = 2sin(3x)cos(3x)(3) = 6sin(3x)cos(3x) = 0

sin(3x) = 0 when 3x = 0, ±π, ±2π, ±3π, ... So, x = 0, ±π/3, ±2π/3, ...= kπ/3, where k is any integer.

cos(3x) = 0 when 3x = π/2 + kπ. So, x = π/6 + kπ/3.