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

May 5, 2017

There are no critical numbers (maximums or minimums)

#### Explanation:

Critical numbers are found when $f ' \left(x\right) = 0$.

Find $f ' \left(x\right)$ using the Quotient Rule: $\left(\frac{u}{v}\right) ' = \frac{v u ' - u v '}{v} ^ 2$

Given: $f \left(x\right) = \frac{x - 1}{x + 3}$

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

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

$f ' \left(x\right) = \frac{\left(x + 3\right) \left(1\right) - \left(x - 1\right) \left(1\right)}{x + 3} ^ 2$

Simplify:

$f ' \left(x\right) = \frac{x + 3 - x + 1}{x + 3} ^ 2 = \frac{4}{x + 3} ^ 2$

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

$\frac{4}{x + 3} ^ 2 = 0$

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]}
{"ft_lang_label":"__label__en","ft_lang_prob":0.6365049,"math_prob":1.0000095,"size":588,"snap":"2021-31-2021-39","text_gpt3_token_len":163,"char_repetition_ratio":0.15582192,"word_repetition_ratio":0.06451613,"special_character_ratio":0.2840136,"punctuation_ratio":0.12820514,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000085,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-25T10:14:28Z\",\"WARC-Record-ID\":\"<urn:uuid:bbd319ef-a899-478e-b769-a5fdc9673e97>\",\"Content-Length\":\"34641\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4e848892-309e-4556-b6c7-61bbae1e1258>\",\"WARC-Concurrent-To\":\"<urn:uuid:4a70ff1a-ffa6-4a49-8727-3484a63b4ecb>\",\"WARC-IP-Address\":\"216.239.38.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-find-the-critical-numbers-for-f-x-x-1-x-3-to-determine-the-maximum-an\",\"WARC-Payload-Digest\":\"sha1:GSCRTQ6I4G4NJTJBVLPSFCLFNMNULY2A\",\"WARC-Block-Digest\":\"sha1:EF3PZ6RQQHBJMBTO6F64OFOJGVRE32DA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057615.3_warc_CC-MAIN-20210925082018-20210925112018-00369.warc.gz\"}"}