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

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

Answer 2

Answer:

$x=\dfrac{\pi}{4},-\dfrac{\pi}{4},\dfrac{3\pi}{4},-\dfrac{\pi}{4}$

Step-by-step 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

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

$\tan^2x=1$

$\tan x=\pm 1$

$x=\dfrac{\pi}{4},-\dfrac{\pi}{4},\dfrac{3\pi}{4},-\dfrac{\pi}{4}$

Horizontal tangents are,

$y=\dfrac{1}{2}\ and \ y=-\dfrac{1}{2}$