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Question
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
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Solution
We have f(x) = x5 – 5x4 + 5x3 – 1
⇒ f '(x) = 5x4 – 20x3 + 15x2
For local maxima and local minima, f '(x) = 0
⇒ 5x4 – 20x3 + 15x2 = 0
⇒ 5x2(x2 – 4x + 3) = 0
⇒ 5x2(x2 – 3x – x + 3) = 0
⇒ x2(x – 3)(x – 1) = 0
∴ x = 0, x = 1 and x = 3
Now f '(x) = 20x3 – 60x2 + 30x
⇒ `"f''"(x)_("at" x = 0)` = 20(0)3 – 60(0)2 + 30(0) = 0
Which is neither maxima nor minima.
∴ f (x) has the point of inflection at x = 0
`"f''"(x)_("at" x = 1)` = 20(1)3 – 60(1)2 + 30(1)
= 20 – 60 + 30
= –10 < 0 Maxima
`"f''"(x)_("at" x = 2)` = 20(3)3 – 60(3)2 + 30(3)
= 540 – 540 + 90
= 90 > 0 Minima
The maximum value of the function at x = 1
f (x) = (1)5 – 5(1)4 + 5(1)3 – 1
= 1 – 5 + 5 – 1
= 0
The minimum value at x = 3 is
f (x) = (3)5 – 5(3)4 + 5(3)3 – 1
= 243 – 405 + 135 – 1
= 378 – 406
= – 28
Hence, the function has its maxima at x = 1 and the maximum value = 0 and it has minimum value at x = 3 and its minimum value is – 28.
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