Question

If(x) = x+$\frac{1}{x}$, x > 0, then its greatest value is

(a) $-$2

(b) 0

(c) 3

(d) none of these

Answer 1

(d) none of these

$$\begin{gathered} \mathrm{Given}:f\left(x\right)=x+\frac{1}{x} \\ \Rightarrow f\text{'}\left(x\right)=1-\frac{1}{x^2} \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow 1-\frac{1}{x^2}=0 \\ \Rightarrow x^2-1=0 \\ \Rightarrow x^2=1 \\ \Rightarrow x=\pm 1 \\ \Rightarrow x=1\left(\mathrm{Given}:\text{x>0}\right) \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=\frac{2}{x^3} \\ \Rightarrow f\text{'}\text{'}\left(1\right)=2>0 \\ \mathrm{So},x=1\mathrm{is}\mathrm{a}\mathrm{local}\mathrm{minima}. \end{gathered}$$