Question

Find the maximum and minimum of the following functions:

f(x) = x³ – 9x² + 24x

Answer 1

f(x) = x³ – 9x² + 24x

∴ the roots of f'(x) = 0 are x₁ = 2 and x₂ = 4.

(a) f”(2) = 6(2) – 18 = -6 < 0

∴ by the second derivative test, f has maximum at x = 2 and maximum value of f at x = 2

(b) f”(4) = 6(4) – 18 = 6 > 0

∴ by the second derivative test, f has minimum at x = 4 and minimum value of f at x = 4

Hence, the function f has maximum value 20 at x = 2 and minimum value 16 at x = 4.