if x>0 then minimum value of x³+1/x³ is

Question

Answer 1

we should find the minimum of the \( x^3+\frac{1}{x^3} \).

so we derive from the function and set it to zero:

\( 3x^2-\frac{3}{x^2}=0 \Rightarrow 3x^4-3=0 \Rightarrow 3x^4=3 \Rightarrow x^4=1 \Rightarrow x=\pm 1 \overset{x>0}\Longrightarrow x=1 \)

\( \mathrm{min}(x^3+\frac{1}{x^3})=1+\frac{1}{1^3}=1+1=2 \)