Maximum Value?

Question

What is the maximum value of sin(x) + cos(x)?
I have differentiated and set equal to 0 so I know sin(x)=cos(x)
Not sure how I get from this to the answer though?
Thanks for any help.

Answer 1

you advanced a lot.From sin(x) = cos(x) you get tg(x)=1
(dividing) by cos(x).so x in the first cuadrant where sin and cos are positive x=45 degrees.
cos 45 +sen45 = sqrt2/2 +sqrt2/2 = sqrt2

Answer 2

Over the range of 0° to 90° the value of Sin will go from 0 to 1.
Over the range of 0° to 90° the value of Cos will go from 1 to 0.
So what value for x, will give a maximum?
Try 45°.

Sin x + Cos x =
Sin 45° + Cos 45° =
0∙707 106 781... + 0∙707 106 781... = 1∙414 213 562..
It's the intersecting point of the two curves.

Answer 3

Divide both sides by cos x where x is not allowed to equal pi/2 and get tan x = 1 or x = pi/4 + N2pi

Answer 4

i know this is quite simple,but i love this soloution:
we have:

(ac + bd)^2 =< (a^2 + b^2)(c^2 + d^2)

for all real a,b,c,d.

so,assume:
a = sin(x)
b = cos(x)
c = 1
d = 1
then :

( sin(x) + cos(x) )^2 =< ( sin^2(x) + cos^2(x) )( 1 + 1 )

=> ( sin(x) + cos(x) )^2 =< ( 1 ) (2)
=> - ( 2^(1/2) ) =< sin(x) + cos(x) =< 2^(1/2)
so the minimum value is - ( 2^(1/2) )
and the maximum value is 2^(1/2)

was'nt it lovely? :)

Answer 5

sin(x) = cos(x)
sin(x) / cos(x) = 1
tan(x) = 1
x = atan(1) = pi / 4

all the solutions: pi/4 + k pi , where k is an integer

Answer 6

f(x)=sinx+cosx
f'(x)=cosx-sinx
setting it tozero
sinx=cosx
this is true for x= pi/4
sinpi/4+cospi/4
1/rt2+1/rt2
=2/rt2
=rt2