Find the minimum value of #f(x)=(x^2+1/x)/(x^2-(1-1/x^2)/(1/x+1/x^2)# over the interval #1lexle2#. Write your answer as an exact decimal?

1 Answer
Dec 17, 2017

#f(2) = 1.5#

Explanation:

#f(x) = (x^2+1/x)/(x^2-(1-1/x^2)/(1/x+1/x^2))#

#color(white)(f(x)) = (x^2+1/x)/(x^2-(x^2-1)/(x+1)#

#color(white)(f(x)) = (x^2+1/x)/(x^2-(x-1))#

#color(white)(f(x)) = (x^2+1/x)/(x^2-x+1)#

#color(white)(f(x)) = (x^3+1)/(x(x^2-x+1))#

#color(white)(f(x)) = ((x+1)(x^2-x+1))/(x(x^2-x+1))#

#color(white)(f(x)) = (x+1)/x#

#color(white)(f(x)) = 1+1/x#

This function is monotonically decreasing over the interval #[1, 2]#

So the minimum value occurs when #x=2#, namely #f(2) = 1+1/2 = 3/2 = 1.5#