How do you find the critical points of f(x)=sinx+cosxf(x)=sinx+cosx?

1 Answer

y = sin x + cos xy=sinx+cosx
Use the Trig Identity sin + cos x = sqrt{2} sin (x + pi/4)sin+cosx=2sin(x+π4).
y = sqrt{2} sin (x + pi/4)y=2sin(x+π4)
yy min when sin (x + pi/4) = -1 rArr x + pi/4 = 3/2 pi rArr x = 5/4 pisin(x+π4)=1x+π4=32πx=54π.
yy max when sin(x + pi/4) = 1 rArr x + pi/4 = sin pi/2 rArr x = pi/4sin(x+π4)=1x+π4=sinπ2x=π4.
In the interval (0, 2 pi)(0,2π) there are 22 answers: pi/4π4 and 5/4 pi54π.
Check
When x = pi/4 rArr y = sqrt{2}/2 + sqrt{2}/2 = sqrt{2}x=π4y=22+22=2 (Max)
When x = 5/4 pi rArr y = -sqrt{2}/2 - sqrt{2}/2 = - sqrt{2}x=54πy=2222=2 (Min)