x5-5x4+5x3+10 maxima and minima
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Step-by-step explanation:
To obtain the absolute maxima or minima for the function f(x)
Find f
′
(x) and put f
′
(x)=0
Given
x
5
−5x
4
+5x
3
−10
f(x)=x
5
−5x
4
+5x
3
−10
f(x)=x
5
−5x
4
+5x
3
−10
dx
df(x)
=f
′
(x)=5x
4
−20x
3
+15x
2
5x
4
−20x
3
+15x
2
=0
5x
2
(x−3)(x−1)=0
x=0, x=1, x=3
To find the maximum find f
′′
(x) and substitute the value of x
dx
df
′
(x)
=f
′′
(x)=20x
3
−60x
2
+30x
When, x=3
f
′′
(x)=20x
3
−60x
2
+30x=20(3)
3
−60(3)
2
+30(3)=90 >0
When, x=0
f
′′
(x)=20x
3
−60x
2
+30x=20(0)
3
−60(0)
2
+30(0)=0
When, x=1
f
′′
(x)=20x
3
−60x
2
+30x=20(1)
3
−60(1)
2
+30(1)=−10 <0
∴ The given function has the maximum value when x=3
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