Question

Please locate the inflection points and relative extrema of y=x-sinx on the interval [0,4pi] using the second derivative test.

Answer 1

find the 1st and second derivatives

\[\frac{dy}{dx} = 1 - \cos(x)\]

\[\frac{d^2y}{dx^2} = \sin(x)\]

set \[\frac{dy}{dx} = 0\] to find stationary points

then cos(x) = 1 x = 0, 2 pi, 4pi

set \[\frac{d^2y}{dx^2} = 0\]

sin(x) = 0 so x = 0, pi, 2pi, 3pi, 4pi.

test the stationary points from the 1st derivative in the 2nd derivative

sin(0) = 0 sin(2pi) = 0 sin(4pi) = 0

then the points where x = 0, 2pi, 4pi are horizontal points of inflexion.

test either side of the points for a change in the sign of the 2nd derivative



then the points of inflexion are \[(0,0)**,...(\pi, \pi)...(2\pi, 2\pi)**, .... (3\pi, 3\pi),.... (4\pi, 4\pi)**\]

and ** are horizontal points of inflexion