# Based off of this information, what conclusions can be made about Rolle’s Theorem?

f(x) = 1 − x^(2/3)

Recall that f must be continuous on [a, b], differentiable on (a, b), and f(a) = f(b) in order to apply Rolle’s Theorem.
——
Here, a = -1, b = 1, and f(x) = 1 – x^(2/3).

We know that f(-1) = f(1) = 0, and f is continuous on [-1, 1] (since the exponent is not negative and cube roots are defined for all x).

However, you found that f ‘(c) = 0 has no solutions (in (-1, 1)).
Since Rolle’s Theorem does not apply, the only hypothesis of the theorem that can not be satisfied is “f is not differentiable on (−1, 1)”.

This is indeed the case, because f ‘(x) = (2/3)x^(-1/3) is not defined at x = 0. So, (D) is the answer.

I hope this helps!
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://chargersgame.org/based-off-of-this-information-what-conclusions-can-be-made-about-rolles-theorem/", "fetch_time": 1670407230000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2022-49/segments/1669446711151.22/warc/CC-MAIN-20221207085208-20221207115208-00364.warc.gz", "warc_record_offset": 200757756, "warc_record_length": 16123, "token_count": 251, "char_count": 744, "score": 3.71875, "int_score": 4, "crawl": "CC-MAIN-2022-49", "snapshot_type": "latest", "language": "en", "language_score": 0.9574073553085327, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00009-of-00064.parquet"}