Question

Let f(x) = x³+3x²$-$9x+2. Then, f(x) has

(a) a maximum at x = 1
(b) a minimum at x = 1
(c) neither a maximum nor a minimum at x = $-$3
(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{b}\right)\text{a minimum at x=1} \\ \mathrm{Given}:f\left(x\right)=x^3+3x^2-9x+2 \\ \Rightarrow f\text{'}\left(x\right)=3x^2+6x-9 \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow 3x^2+6x-9=0 \\ \Rightarrow x^2+2x-3=0 \\ \Rightarrow \left(x+3\right)\left(x-1\right)=0 \\ \Rightarrow x=-3,1 \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=6x+6 \\ \Rightarrow f\text{'}\text{'}\left(1\right)=6+6=12>0 \\ \mathrm{So},x=1\mathrm{is}\mathrm{a}\mathrm{local}\text{mini}\mathrm{ma}. \\ \mathrm{Also}, \\ f\text{'}\text{'}\left(-3\right)=-18+6=-12<0 \\ \mathrm{So},x=-3\mathrm{is}\mathrm{a}\mathrm{local}\mathrm{m}\text{axi}\mathrm{ma}. \end{gathered}$$