Question

What is the minimum value of `g(x) = (x-1)/(x^2+4)?` on the interval `[-2,2]`?

Answer 1

Minimum value is at `x=1-sqrt 5 approx "-"1.236`;

`g(1 - sqrt 5) = -(1+ sqrt 5)/(8) approx "-"0.405`.

Explanation:

On a closed interval, the possible locations for a minimum will be:

• a local minimum inside the interval, or
• the endpoints of the interval.

We therefore compute and compare values for `g(x)` at any `x in ["-2",2]` that makes `g'(x)=0`, as well as at `x="-2"` and `x=2`.

First: what is `g'(x)`? Using the quotient rule, we get:

`g'(x)=((1)(x^2+4)-(x-1)(2x))/(x^2+4)^2`
`color(white)(g'(x))=(x^2+4-2x^2+2x)/(x^2+4)^2`
`color(white)(g'(x))=-(x^2-2x-4)/(x^2+4)^2`

This will equal zero when the numerator is zero. By the quadratic formula, we get

`x^2-2x-4=0" "=>" "x=1+-sqrt 5 approx {"-1.236", 3.236}`

Only one of these `x`-values is in `["-2",2]`, and that is `x=1-sqrt 5`.

Now, we compute:

1. `g("-2") = ("-"2-1)/(("-2")^2+4)="-3"/8="-"0.375`

2. `g(1 - sqrt 5) = (1 - sqrt 5 -1)/((1 - sqrt 5)^2+4)=("-"sqrt 5)/(1-2 sqrt 5 + 5+4)`
`color(white)(g(1 - sqrt 5)) = -(sqrt 5)/(10-2sqrt 5) =-(sqrt 5)/((2)(5-sqrt5)) * color(blue)((5+sqrt 5)/(5+ sqrt 5))`
`color(white)(g(1 - sqrt 5)) =-(5+5 sqrt 5)/(2 * (25-5)`
`color(white)(g(1 - sqrt 5)) =-(5(1+sqrt5))/(40)=-(1+sqrt 5)/(8) approx "-"0.405`

3. `g(2)=(2-1)/(2^2+4)=1/8 =0.125`

Comparing these three values of `g(x)`, we see that `g(1-sqrt 5)` is the smallest. So `-(1+ sqrt 5)/8` is our minimum value for `g(x)` on `["-"2, 2]`.