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

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Answer 1 · 2017-05-05T18:08:50

There are no critical numbers (maximums or minimums)

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]}

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