Question

Write the maximum value of f(x) = $x+\frac{1}{x},x>0.$

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}f\left(x\right)=x+\frac{1}{x} \\ \Rightarrow f\text{'}\left(x\right)=1-\frac{1}{x^2} \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow 1-\frac{1}{x^2}=0 \\ \Rightarrow x^2=1 \\ \Rightarrow x=1,-1 \\ \mathrm{But}x<0 \\ \Rightarrow x=-1 \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=\frac{1}{x^3} \\ \mathrm{At}x=-1: \\ f\text{'}\text{'}\left(-1\right)=\frac{2}{{\left(-1\right)}^3}=-2<0 \\ \mathrm{So},x=-1\mathrm{is}\mathrm{a}\mathrm{point}\mathrm{of}\mathrm{local}\mathrm{maximum}. \\ \mathrm{Thus},\mathrm{the}\mathrm{local}\mathrm{maximum}\mathrm{value}\mathrm{is}\mathrm{given}\mathrm{by} \\ f\left(-1\right)=-1+\frac{1}{-1}=-1-1=-2 \end{gathered}$$