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 `1 le x le 2`. Write answer as *exact* decimal?

Answer 1

Minimum value `f(2) = 1.5`

Explanation:

Given:

`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^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`

with excluded value `x != -1`

Note that `1/x` is monotonically decreasing in the interval `[1, 2]`.

So the minimum value is attained when `x=2`:

`f(2) = 1 + 1/2 = 1.5`

Answer 2

`3/2`

Explanation:

`"Multiply numerator and denominator by (1/x + 1/x²)."`
`"Then we obtain f(x) = : "`

`((x^2 + 1/x)(1/x + 1/x^2)) / ((x^2(1/x + 1/x^2) + 1/x^2 - 1)`
`= (x + 1 + 1/x^2 + 1/x^3)/(x + 1 + 1/x^2 - 1)`

`"Now multiply numerator and denominator by x³ : "`

`= (x^4 + x^3 + x + 1)/(x^4 + x)`

`"Now derive using the quotient rule."`

`f'(x) = ((4x^3+3x^2+1)(x^4+x)-(x^4+x^3+x+1)(4x^3+1))/(x^4+x)^2`

`"The derivative is zero when the numerator is zero :"`

`- x^6 - 2 x^3 - 1 = 0`
`=> (x^3 + 1)^2 = 0`
`=> x^3 + 1 = 0`
`=> x = -1`

`"There is a problem though as the denominator is also zero"`
`"for this value of x, so we have the case 0/0, so we must"`
`"divide away the common factor (x+1)^2 :"`

`f'(x) = -(x^3+1)^2 / (x^2(x^3+1)^2)`
`= -1/x^2`

`"So the derivative is always negative. This means that the"`
`"function is ever decreasing, so the minimum over the interval"`
`"[1, 2] is reached for x=2 :"`

`f(2) = (2^4 + 2^3 + 2 + 1)/(2^4 + 2)`
`= 27/18`
`= 3/2`

Answer 3

Minimum value is `1.5`

Explanation:

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

= `((x^3+1)/x)/(x^2-(x^2(1-1/x^2))/(x^2(1/x+1/x^2))`

= `((x^3+1)/x)/(x^2-(x^2-1)/(x+1)`

= `((x^3+1)/x)/(x^2-(x-1)`

= `((x+1)(x^2-x+1))/x xx1/(x^2-x+1)`

= `(x+1)/x=1+1/x`

Observe that at `x=1` we have `f(x)=2` and as `x` increases to `2`, the value of `1/x` comes down

and it is minimum wheen `x=2` and it is `1+1/2=1.5`

graph{(x^2+1/x)/(x^2-(1-1/x^2)/(1/x+1/x^2) [-0.983, 4.017, 0.23, 2.73]}