\(f(x) = x^3 - 9x^2 + 24x\)
\(f'(x) = 3x^2 - 18x + 24\)
\(f'(x) = 0 \)
⇒ \(3x^2 - 18 x + 24 = 0\)
⇒ \(x^2 - 6x + 8 = 0\)
⇒ \((x -2) (x -4) = 0\)
⇒ \(x = 2\;or\;x = 4\)
\(f''(x) = 6x - 18\)
\(f''(2) = 12 - 18 < 0\)
\(f''(4) = 24 - 18 > 0\)
\(\therefore x = 2 \) is point of maxima and \(x = 4\) is point of minima.
Local maximum value \(= f(2) \)
\(= 8 - 36 + 48 \)
\(= 20\)
Local minimum value \(= f(4) \)
\(= 64 - 144 + 96 \)
\(= 160 - 144\)
\(= 16\)
