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

1 Answer
May 5, 2017

Answer:

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]}