If $f(x) = x - 3{x^{\frac{2}{3}}},{\text{ }}x > 0$ ,

(a)     find the x-coordinate of the point P where $f'(x) = 0$ ;

(b)     determine whether P is a maximum or minimum point.

(a)     $f'(x) = 1 - \frac{2}{{{x^{\frac{1}{3}}}}}$     A1

$\Rightarrow 1 - \frac{2}{{{x^{\frac{1}{3}}}}} = 0 \Rightarrow {x^{\frac{1}{3}}} = 2 \Rightarrow x = 8$     A1

(b)     $f''(x) = \frac{2}{{3{x^{\frac{4}{3}}}}}$     A1

$f''(8) > 0 \Rightarrow {\text{ at }}x = 8,{\text{ }}f(x){\text{ has a minimum.}}$     M1A1

[5 marks]

Most candidates were able to correctly differentiate the function and find the point where $f'(x) = 0$ . They were less successful in determining the nature of the point.