Question

The least and greatest values of f(x) = x³$-$6x²+9x in [0,6], are

(a) 3,4

(b) 0,6

(c) 0,3

(d) 3,6

Answer 1

$$\begin{gathered} \mathrm{Given}:f\left(x\right)=x^3-6x^2+9x \\ \Rightarrow f\text{'}\left(x\right)=3x^2-12x+9 \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow 3x^2-12x+9=0 \\ \Rightarrow x^2-4x+3=0 \\ \Rightarrow \left(x-1\right)\left(x-3\right)=0 \\ \Rightarrow x=1,3 \\ \mathrm{Now}, \\ f\left(0\right)=0^3-6{\left(0\right)}^2+9\left(0\right)=0 \\ f\left(1\right)=1^3-6{\left(1\right)}^2+9\left(1\right)=1-6+9=4 \\ f\left(3\right)=3^3-6{\left(3\right)}^2+9\left(3\right)=27-54+27=0 \\ f\left(6\right)=6^3-6{\left(6\right)}^2+9\left(6\right)=216-216+54=54 \end{gathered}$$

The least and greatest values of f(x) = x³- 6x²+9x 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.