identify the coordinates of any local and absolute extreme points and inflection points. Graph the function.

Question

y = sin x cos x, 0≤x≤π

Answer 1

y=cosxsinx

use a graphing calculator

max 1/2, min -1/2

where

x=pi/4+npi=5npi/4

(5npi/4, +/-1/2)

inflection points= x intercepts, x=0, pi/2

y=0

x=npi, pi/2+npi=3npi/4

(npi, 0), 3npi/4,0)

y'= sinx(-sinx)+cosxcosx= cos2x=0

2x = cos^-1(0) = pi/2, 3pi/2

x = pi/4+npi, 3pi/4 + npi where n=any integer

y=sin(pi/4)cos(pi/4) = (sqr2/2)^2 = 2/4= 1/2

y=sin(3pi/4)cos(3pi/4) = 0

local & global max are (5npi/4, +/-1/2),

y'=cos^2(x)-sin^2(x)

y"= 2cosx(-sinx)-2sinx(cosx)=0

cosxsinx(-4)=0

cosx=0=sinx

= 2npi, 3npi are the inflection points, in between the max and min points

(3npi/2, 0)

y'=cos2x

y"=-2sin2x=0

2x=0,pi

x=0, pi/2

+npi

Answer 2

Use the Product Rule to determine first and second derivatives.