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By completing the following activity, examine the function f(x) = x^{3} – 9x^{2} + 24x for maxima and minima
Solution: f(x) = x^{3} – 9x^{2} + 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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Solution
f(x) = x^{3} – 9x^{2} + 24x
∴ f'(x) = 3x^{2} – 18x + 24
∴ f''(x) = 6x – 18
For extreme values, f'(x) = 0, we get
3x^{2} – 18x + 24
∴ x^{2} – 6x + 8 = 0
∴ x^{2} – 4x – 2x + 8 = 0
∴ x(x – 4) – 2(x – 4) = 0
∴ (x – 4)(x – 2) = 0
x = 2 or 4
∴ f''(2) = 6(2) – 18
= 12 – 18
= – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = f(2)
= (2)^{3} – 9(2)^{2} + 24(2)
= 8 – 36 + 48
= 20
∴ f''(4) = 6(4) – 18
= 24 – 18
= 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = f(4)
= (4)^{3} – 9(4)^{2} + 24(4)
= 64 – 144 + 96
= 16
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