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Byju's Answer
Standard XII
Mathematics
Direction Cosines of a Line Passing through Two Points
Determine the...
Question
Determine the values of x for which the function f(x) = x
2
− 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x
2
− 6x + 9 where the normal is parallel to the line y = x + 5.
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Solution
Here
,
f
x
=
x
2
-
6
x
+
9
f
'
x
=
2
x
-
6
For
f
(
x
) to be increasing, we must have
f
'
x
>
0
⇒
2
x
-
6
>
0
⇒
2
x
>
6
⇒
x
>
3
⇒
x
∈
3
,
∞
So,
f
(
x
)
is increasing on
3
,
∞
.
For
f
(
x
) to be decreasing, we must have
f
'
x
<
0
⇒
2
x
-
6
<
0
⇒
2
x
<
6
⇒
x
<
3
⇒
x
∈
-
∞
,
3
So,
f
(
x
)
is decreasing on
-
∞
,
3
.
Let (x, y) be the coordinates on the given curve where the normal to the curve is parallel to the given line.
Slope of the given line = 1
Slope of tangent =
d
y
d
x
x
,
y
=
2
x
-
6
Slope of normal =
-
1
Slope of tangent
=
-
1
2
x
-
6
Now
,
Slope of normal = Slope of the given line
-
1
2
x
-
6
=
1
-
1
=
2
x
-
6
2
x
=
5
x
=
5
2
Given curve is
y
=
x
2
-
6
x
+
9
=
25
4
-
15
+
9
=
1
4
x
,
y
=
5
2
,
1
4
Hence
,
the
coordinates
are
5
2
,
1
4
.
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0
Similar questions
Q.
The normal line to a given curve at each point
(
x
,
y
)
on the curve passes through the point
(
3
,
0
)
. If the curve contains the point
(
3
,
4
)
, then the equation of the curve is
Q.
If
(
x
2
+
1
)
(
y
2
+
1
)
+
9
=
6
(
x
+
y
)
(where
x
,
y
∈
R
), then the value of
(
x
2
+
y
2
)
is,
Q.
Consider the function
y
=
x
2
−
6
x
+
9
. the maximum value of
y
obtained when
x
varies over the interval
2
to
5
is
Q.
Area common to the curve
y
=
√
9
−
x
2
and
x
2
+
y
2
=
6
x
is
9
√
3
2
+
n
π
. Find
n
Q.
Find the interval in which the following function is increasing or decreasing.
f
(
x
)
=
x
4
4
+
2
3
x
3
−
5
2
x
2
−
6
x
+
7
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