Maxima and
minima
x3 (x-2)3
Answers
Answered by
1
Answer:
maxima at
(
−
4
,
83
)
minima at
(
2
,
−
25
)
Explanation:
We have:
f
'
(
x
)
=
3
x
2
+
6
x
−
24
At a max/min (turning point)
f
'
(
x
)
=
0
⇒
3
x
2
+
6
x
−
24
=
0
∴
x
2
+
2
x
−
8
=
0
∴
(
x
−
2
)
(
x
+
4
)
=
0
∴
x
=
−
4
,
2
,
When
x
=
−
4
⇒
f
(
−
4
)
=
−
64
+
48
+
96
+
3
=
83
When
x
=
2
⇒
f
(
2
)
=
8
+
12
−
48
+
3
=
−
25
To determine the nature of the turning points we look at the second derivative. Differentiating again wrt
x
:
f
'
'
(
x
)
=
6
x
+
6
When
x
=
−
4
⇒
f
'
'
(
−
4
)
<
0
⇒
maximum
When
x
=
2
⇒
f
'
'
(
2
)
>
0
⇒
minimum
So the maxima and minima are:
maxima at
(
−
4
,
83
)
minima at
(
2
,
−
25
)
We can confirm these results graphically:
graph{x^3+3x^2-24x+3 [-10, 10, -50, 100.]}
Similar questions
Math,
3 months ago
Physics,
3 months ago
Social Sciences,
8 months ago
Math,
8 months ago
Science,
1 year ago