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(a) Find all points on the graph of ff where the tangent line is horizontal.

(b)  Graph the function and the horizontal tangent lines on the interval [−2π,2π][−2π,2π] on the same screen.

f(x)=2sinx+cosx


I honestly have no idea how to do this..the answer looks crazy too. Any help is appreciated.
 

 Feb 21, 2016
 #1
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f(x)=2sinx+cosx

 

1st....find the derivative

 

f ' (x)  = 2cosx  - sinx

 

Set this =  0........the solutions will tell us where the derivative  = 0, that is, where the tangent lines are horizontal

 

2cosx - sinx  = 0      square both sides.......this may produce some "extraneous" solutions......we'll deal wth this possibility in a sec

 

4cos^2x = sin^2x    and notce that we can write

 

4cos^2x  = 1 - cos^2x       add cos^2x to both sides

 

5cos^2x  =  1        divide both sides by 5

 

cos^2x =  1/5       take the square root of both sides

 

cosx  = +1/ √5        or cos x  = -1 / √5

 

Let's deal with the first possibility......I'm going to work with degrees....we'll covert back to radians

 

cosx  = +1/ √5      and this happens at about  63.4° and at about 296.56°

 

And the second solution is

 

cosx  = -1/ √5    and this happens at about about 116.56°  and at about 243.43°

 

Now......look at the graph [in degrees] : https://www.desmos.com/calculator/dokwgbq8vg

 

Notice that the only two places  from  [0 to 360] that the tangent lines are horizontal occur at about  63.4°  and at about 243.43°

 

So the other two solutions are  extraneous [ produced because we squared the original equation]

 

And the general solutions on [0, 360] degrees  are

 

x = 63.4°  + n 360°     and x = 243.43° + n 360°   where n is an integer

 

So.....converting to radians we have

 

x = 1.1065 + n(2pi)     and x = 4.248 +n (2pi)   where n is an integer

 

So......the correct answers  in radian measure on [0, 2pi]  are

 

x = 1.1065 rads  and x = 4.248 rads  on [ 0, 2p]

 

And ....since the answer also specified the interval [-2pi, 0], we have

 

x = 1.1065 - 2pi  = -5.177rads     and  x = 4.248 - 2pi  = -2.034 rads  on [-2pi, 0]

 

Here's the graph in rads : https://www.desmos.com/calculator/apqa44zvad

 

 

cool cool cool

 Feb 21, 2016

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