Question

How do you find the critical numbers for `f(x) = (x - 1 )/( x + 3)` to determine the maximum and minimum?

Answer 1

There are no critical numbers (maximums or minimums)

Explanation:

Critical numbers are found when `f'(x) = 0`.

Find `f'(x)` using the Quotient Rule: `(u/v)' = (v u' - u v')/v^2`

Given: `f(x) = (x-1)/(x+3)`

Let `u = x-1; " " u' = 1`

Let `v = x + 3; " " v' = 1`

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

Simplify:

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

Find critical numbers `(f'(x) = 0`:

`4/(x+3)^2 = 0`

Multiply both sides by the denominator: `4 = 0`

There are no critical numbers (maximums or minimums).

This can be seen by graphing the function:

graph{(x-1)/(x+3) [-11.66, 8.34, -5.12, 4.88]}