Question

Find the values of x on the interval [−π, π] where the tangent line to the graph of y = sin(x) cos(x) is horizontal. (Enter your answers as a comma-separated list.)

Answer 1

`x=(\pi)/(4),-(\pi)/(4),(3\pi)/(4),-(\pi)/(4)`

Explanation:

Function: y= sin(x) cos(x)

To find the value of x where tangent line is horizontal on the interval [−π, π]

Slope = y'

`y=\sin x\cos x`

derivative of y

`y'=\cos^2x-\sin^2x`

For horizontal tangent line, slope must be 0

`\cos^2x-\sin^2x=0`

`\tan^2x=1`

`\tan x=\pm 1`

`x=(\pi)/(4),-(\pi)/(4),(3\pi)/(4),-(\pi)/(4)`

Horizontal tangents are,

`y=(1)/(2)\ and \ y=-(1)/(2)`

Answer 2

A trig identity is asinucosu=a/2sin(2u)So you can write your equation asy=sin(x)cos(x)=1/2sin(2x)Use the crain rule herey′=d/dx1/2sin(2x)=1/2cos(2x)d/dx2x=cos(2x)The curve will have horizontal tangents when y' = 0.y′=0=cos(2x)On the interval [-pi, pi], solution to that isx=±π4,±3π4