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

1 Answer
Nghi N. · Maurizio Giaffredo
Apr 22, 2015

#y = sin x + cos x#
Use the Trig Identity #sin + cos x = sqrt{2} sin (x + pi/4)#.
#y = sqrt{2} sin (x + pi/4)#
#y# min when #sin (x + pi/4) = -1 rArr x + pi/4 = 3/2 pi rArr x = 5/4 pi#.
#y# max when #sin(x + pi/4) = 1 rArr x + pi/4 = sin pi/2 rArr x = pi/4#.
In the interval #(0, 2 pi)# there are #2# answers: #pi/4# and #5/4 pi#.
Check
When #x = pi/4 rArr y = sqrt{2}/2 + sqrt{2}/2 = sqrt{2}# (Max)
When #x = 5/4 pi rArr y = -sqrt{2}/2 - sqrt{2}/2 = - sqrt{2}# (Min)

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