SOLUTION: If f(x)=sinx+cos2x,determine which sub-intervals of (0,2pi) f(x) is increasing on and which intervals f(x) is decreasing.

Question 1067370: If f(x)=sinx+cos2x,determine which sub-intervals of (0,2pi)
f(x) is increasing on and which intervals f(x) is decreasing.
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!


If f(x)=sinx+cos2x,determine which sub-intervals of (0,2pi)
f(x) is increasing on and which intervals f(x) is decreasing.

f(x) = sin(x) + cos(2x)
f'(x) = cos(x) - 2sin(2x)

We set f'(x)=0 to see if there are any critical
values as far as increasing or decreasing:

cos(x) - 2sin(2x) = 0

cos(x) -2[2sin(x)cos(x)] = 0

cos(x) -4sin(x)cos(x) = 0

cos(x)[1 - 4sin(x)] = 0

cos(x) = 0;   1 - 4sin(x) = 0
x = ; -4sin(x) = -1
                                        sin(x) = 1/4
                                             x = .2527, 2.8889

Substituting test values on each interval, we find:

It's increasing on 
It's decreasing on 
It's increasing on  
It's decreasing on 
It's increasing on 



The 1st green circle on the graph is where x = 0.2527
The 2nd green circle on the graph is where x = 
The 3rd green circle on the graph is where x = 2.8889
The 4th green circle on the graph is where x = 

Edwin

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