#### Question

f(x) = x^{3} (x \[-\] 1)^{2 }.

#### Solution

\[\text { Given }: f\left( x \right) = x^3 \left( x - 1 \right)^2 \]

\[ \Rightarrow f'\left( x \right) = 3 x^2 \left( x - 1 \right)^2 + 2 x^3 \left( x - 1 \right)\]

\[\text { For a local maximum or a local minimum, we must have }\]

\[ f'\left( x \right) = 0\]

\[ \Rightarrow 3 x^2 \left( x - 1 \right)^2 + 2 x^3 \left( x - 1 \right) = 0\]

\[ \Rightarrow x^2 \left( x - 1 \right)\left\{ 3x - 3 + 2x \right\} = 0\]

\[ \Rightarrow x^2 \left( x - 1 \right)\left( 5x - 3 \right) = 0\]

\[ \Rightarrow x = 0, 1, \frac{3}{5}\]

Since f '(x) changes from negative to positive when x increases through 1, x = 1 is the point of local minima.

The local minimum value of f (x) at x = 1 is given by \[\left( 1 \right)^3 \left( 1 - 1 \right)^2 = 0\]

Since f '(x) changes from positive to negative when x increases through \[\frac{3}{5}\], *x* = \[\frac{3}{5}\] is the point of local maxima.

#### Notes

The solution in the book is incorrect. The solution here is created according to the question given in the book.