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Question
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
Solution
f(x) = x3 - 9x2 + 24x
∴ f '(x) = 3x2 - 18x + 24
∴ f ''(x) = 6x - 18
Consider, f '(x) = 0
∴ 3x2 - 18x + 24 = 0
∴ 3(x2 - 6x + 8) = 0
∴ 3(x - 4)(x - 2) = 0
∴ (x - 4)(x - 2) = 0
∴ x = 2 or x = 4
For x = 4,
f ''(4) = 6(4) - 18 = 24 - 18 = 6 > 0
∴ f(x) is minimum at x = 4
∴ Minima = f(4) = (4)3 - 9(4)2 + 24(4)
= 64 - 144 + 96 = 16
For x = 2,
f ''(2) = 6(2) - 18 = 12 - 18 = - 6 < 0
∴ f(x) is maximum at x = 2
∴ Maxima = f(2) = (2)3 - 9(2)2 + 24(2) = 8 - 36 + 48 = 20
∴ Maxima = 20 and Minima = 16
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By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
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