Question

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

Answer 1

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