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.