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Byju's Answer
Standard XII
Mathematics
Second Derivative Test for Local Maximum
Find the maxi...
Question
Find the maximum slope of the curve y=
-
x
3
+
3
x
2
+
2
x
-
27
.
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Solution
Given
:
y
=
-
x
3
+
3
x
2
+
2
x
-
27
.
.
.
1
Slope
=
d
y
d
x
=
-
3
x
2
+
6
x
+
2
Now
,
M
=
-
3
x
2
+
6
x
+
2
⇒
d
M
d
x
=
-
6
x
+
6
For
maximum
or
minimum
values
of
M
,
we
must
have
d
M
d
x
=
0
⇒
-
6
x
+
6
=
0
⇒
6
x
=
6
⇒
x
=
1
Substituing
the
value
of
x
in
eq
.
1
,
we
get
y
=
-
1
3
+
3
×
1
2
+
2
×
1
-
27
=
-
23
d
2
M
d
x
2
=
-
6
<
0
So
,
the
slope
is
maximum
when
x
=
1
and
y
=
-
23
.
∴
At
1
,
-
23
:
Maximum
slope
=
-
3
1
2
+
6
1
+
2
=
-
3
+
6
+
2
=
5
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