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Examine the Function for Maximum and Minimum F(X) = X3 − 9x2 + 24x. - Mathematics and Statistics

Examine the function for maximum and minimum f(x) = x3 − 9x2 + 24x.

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Solution

f(x) = x3 − 9x2 + 24x

∴ f ′(x) = 3x2 − 18x + 24

∴ f ′′(x) = 6x − 18

Now, f ′(x) = 0

∴ 3x2 − 18x + 24 = 0

∴ x2 − 6x + 8 = 0

∴ (x – 4)(x – 2) = 0

∴ x = 2 or x = 4

For x = 2,

f ′′(2) = 6(2) − 18 = 12 − 18 = −6 < 0

∴ f is maximum at x = 2

∴ maximum value = f(2) = (2)3 − 9(2)2 + 24(2) = 8 − 36 + 48 = 20

For x = 4,

f "(4) = 6(4) −18 = 24 −18 = 6 > 0

∴ f is minimum at x = 4

∴ minimum value = f(4) = (4)3 − 9(4)2 + 24(4) = 64 − 144 + 96 = 16

Concept: Maxima and Minima - Introduction of Extrema and Extreme Values
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