Find all points on the graph of the function f(x) = 2 cos(x) + cos2(x) at which the tangent line is horizontal. (Use n as your arbitrary integer.)

Question

Find all points on the graph of the function f(x) = 2 cos(x) + cos²(x)

 at which the tangent line is horizontal. (Use n as your arbitrary integer.)

(x,y)=______ (smaller value)

(x,y)=______ (larger value)

Answer 1

Let me build on Paul M.'s work

sin x + 2 sin x cos x = 0

sin x(1 + 2 cos x) = 0

Paul gives the solutions from the second part of the multiplication, cos x = -1/2

We also have sin x = 0.

180° n for all integers n

Also, Paul mixes degrees and radians for his solution.

120° + 360°n, 240° + 360°n

If we don't want to mix 180 and 360 as the multiplier above, and also easily show the roots in [0°, 360°)

we can write the solution as

(360°n, 120° + 360°n, 180° + 360°n, 240° + 360°n) n an integer

Answer 2

desmos.com/calculator/0ku4fxvmai

Answer 3

At a horizontal tangent f'(x)=0.

f'(x)=-2sin x - 2sin 2x = 0

sin x + 2 sin x cos x = 0

cos x =-1/2

x = 120° and 240° and 2πn plus those 2 angles for integer n