Question

The minimum value of f(x) = $x4-x2-2x+6$ is
(a) 6
(b) 4
(c) 8
(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{b}\right)4 \\ \mathrm{Given}:f\left(x\right)=x^4-x^2-2x+6 \\ \Rightarrow f\text{'}\left(x\right)=4x^3-2x-2 \\ \Rightarrow f\text{'}\left(x\right)=\left(x-1\right)\left(4x^2+4x+2\right) \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow \left(x-1\right)\left(4x^2+4x+2\right)=0 \\ \Rightarrow \left(x-1\right)=0 \\ \Rightarrow x=1 \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=12x^2-2 \\ \Rightarrow f\text{'}\text{'}\left(1\right)=12-2=10>0 \\ \mathrm{So},x=1\mathrm{is}\mathrm{a}\mathrm{local}\text{minima}. \\ \mathrm{The}\mathrm{local}\mathrm{minimum}\mathrm{value}\mathrm{is}\mathrm{given}\mathrm{by} \\ f\left(1\right)=1-1-2+6=4 \end{gathered}$$