Question

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?

Answer 1

`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`