The least and greatest values

Solution:

Given : $f(x)=x^{3}-6 x^{2}+9 x$

$\Rightarrow f^{\prime}(x)=3 x^{2}-12 x+9$

For a local maxima or a local minima, we must have

$f^{\prime}(x)=0$

$\Rightarrow 3 x^{2}-12 x+9=0$

$\Rightarrow x^{2}-4 x+3=0$

$\Rightarrow(x-1)(x-3)=0$

$\Rightarrow x=1,3$

Now,

$f(0)=0^{3}-6(0)^{2}+9(0)=0$

$f(1)=1^{3}-6(1)^{2}+9(1)=1-6+9=4$

$f(3)=3^{3}-6(3)^{2}+9(3)=27-54+27=0$

$f(6)=6^{3}-6(6)^{2}+9(6)=216-216+54=54$

The least and greatest values of $f(x)=x^{3}-6 x^{2}+9 x$ in $[0,6]$ are 0 and 54 , respectively.

Disclaimer: The question in the book has some error. So, none of the options are matching with the solution. The solution here is according to the question given in the book.