Answers and Explanations 255
Answers
301–400
371.
x = 0, π, 2π
Recall that the chain rule states
d
dx
f g x
f g x g x
( )
( ) ( )
(
)= (
) '
'
Begin by finding the derivative of the function f
 
(x) = 2 cos x + sin
2 
x:
= −
+
=
−
x
x
x
x
x
( )
( sin )
sin cos
sin (cos
)
2
2
2
1
x
f '
Then set the derivative equal to zero to find the x values where the function has a
horizontal tangent line:
2
1 0
sin (cos
)
x
x− =
Next, set each factor equal to zero and solve for x: sin x = 0 has the solutions x = 0, π,
2π, and cos x – 1 = 0, or cos x = 1, has the solutions x = 0, 2π. The slope of the tangent
line is zero at each of these x values, so the solutions are x = 0, π, 2π.
372.
6
2
H x
x
( )
[
]
To find the derivative of F x
H x
( )
( )
= [
]
3
, use the chain rule:
= [
]
= [
]
=
[
]
F x
H x
H x
H x
x
H x
x
( )
( )
( )
( )
( )
3
3
2
6
2
2
2
'
'
373.
28
Because the chain rule gives you
= ' (
)
 
  '
f g x
g x
( )
( )
F x
( )
'
, it follows that
= ' (
)
 
  '
= ' −
 
 
=
=
f g
g
f
( )
( )
( )
( ) ( )
( )( )
2
2
2
2
4
7 4
28
F '
374.
–40
Because the chain rule gives you
= ' (
)
 
  '
x
f f x
f x
( )
( )
( )
F '
, it follows that
= ' (
)
 
  '
= ' −
 
  −
=
−
= −
f f
f
f
( )
( )
( )
( ) ( )
( )( )
2
2
2
2
5
8
5
40
F '