Let f(x)=3 x^2-1/x^4+1 (a) At which points does the graph of the f(x) have a horizontal tangent line?
Answers
Answer:
Step-by-step explanation:
The slope of the tangent line of a graph
y
=
f
(
x
)
at a point
x
0
is given by the derivative of
f
at that point, that is,
f
'
(
x
0
)
.
A horizontal tangent line implies a slope of
0
, so our goal is to find the points at which the derivative
f
(
x
)
evaluates to
0
.
Using the quotient rule, we find the derivative as
f
'
(
x
)
=
d
d
x
x
2
x
−
1
=
(
x
−
1
)
(
d
d
x
x
2
)
−
x
2
(
d
d
x
(
x
−
1
)
)
(
x
−
1
)
2
=
2
x
(
x
−
1
)
−
x
2
(
1
)
(
x
−
1
)
2
=
2
x
2
−
2
x
−
x
2
(
x
−
1
)
2
=
x
2
−
2
x
(
x
−
1
)
2
=
x
(
x
−
2
)
(
x
−
1
)
2
Setting this equal to zero, we get
x
(
x
−
2
)
(
x
−
1
)
2
=
0
⇒
x
(
x
−
2
)
=
0
⇒
x
=
0
or
x
=
2
Thus, the graph of
f
(
x
)
has a horizontal tangent line at
OKx
=
0
and
x
=
2
, that is, at thepit (
,
)
4 )
Answer:
Explanation:
First, find the derivative.
f'(x)=3x2+6x+1
