2021-12-17

How do I use the intermediate value theorem to determine whether ${x}^{5}+3{x}^{2}-1=0$ has a solution over the interval $\left[0,3\right]?$ Expert

Explanation:
Bolzanos Joseph Lewis

Expert

Explanation:
According to the intermediate value theorem:
If the function $f\left(x\right)$ is continuous on the interval $\left[a,b\right]$ and $y$ is a number between , then there exists a number $x=c$ in the open interval $\left(a,b\right)$ such that $y=f\left(c\right)$
Here,
$f\left(x\right)={x}^{5}+3{x}^{2}-1$
is a continouos function on the interval $\left[0,3\right]$ as $f\left(x\right)$ is a polynomial function.
$f\left(0\right)=-1$
$f\left(3\right)=243+27-1=269$
$0\in \left(f\left(0\right),f\left(3\right)\right)$
As $f\left(x\right)$ changes sign from $f\left(0\right)$ as $f\left(3\right)$
$\mathrm{\exists }c\in \left(0,3\right)$ such that $f\left(c\right)=0$
The intermediate value theorem is used in this situation. RizerMix

Expert
{"ft_lang_label":"__label__en","ft_lang_prob":0.7858597,"math_prob":1.0000052,"size":572,"snap":"2022-40-2023-06","text_gpt3_token_len":146,"char_repetition_ratio":0.13380282,"word_repetition_ratio":0.0,"special_character_ratio":0.24825175,"punctuation_ratio":0.07,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.999992,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-03T04:42:30Z\",\"WARC-Record-ID\":\"<urn:uuid:b6486af5-7892-458f-93e3-7bfa9c8ae4b6>\",\"Content-Length\":\"278804\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:42718494-6a30-47ba-9223-0ac2c80580e1>\",\"WARC-Concurrent-To\":\"<urn:uuid:1055c16a-f321-415e-804e-8bb6a43fb9dc>\",\"WARC-IP-Address\":\"172.66.43.177\",\"WARC-Target-URI\":\"https://plainmath.net/trigonometry/46909-how-do-i-use-the-intermediate-value-theorem-to-determine\",\"WARC-Payload-Digest\":\"sha1:XPN25ZRGKLJ6WGEPBD7IVI7JFKAGFVVR\",\"WARC-Block-Digest\":\"sha1:3C5ESHLMDLZ4UNAF5HXCQLLVBDVCGAV7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500042.8_warc_CC-MAIN-20230203024018-20230203054018-00326.warc.gz\"}"}