# `f(x) = x^3 - 6x^2 + 15` Find the critical numbers, open intervals on which the function is increasing or decreasing, apply first derivative test to identify all relative extrema.

### Textbook Question

Chapter 3, 3.3 - Problem 22 - Calculus of a Single Variable (10th Edition, Ron Larson).
See all solutions for this textbook.

mathace | (Level 3) Assistant Educator

Posted on

Given: `f(x)=x^3-6x^2+15`

Find the critical values for x by setting the first derivative of the function equal to zero and solving for the x value(s).

`f'(x)=3x^2-12x=0`

`3x(x-4)=0`

`x=0,x=4`

The critical value for the first derivative are x=0 and x=4.

If f'(x)>0, the function is increasing in the interval.

If f'(x)<0, the function is decreasing in the interval.

Choose a value for x that is less than 0.

f'(-1)=15 Since f'(-1)>0 the graph of the function is increasing in the interval (-oo,0).

Choose a value for x that is between 0 and 4.

f'(1)=-9 Since f'(1)<0 the graph of the function is decreasing in the interval

(0, 4).

Choose a value for x that is greater than 4.

f'(5)=15 Since f'(5)>0 the graph of the function is increasing in the interval

(4, `oo).`

Because the function changed direction from increasing to decreasing there will be a relative maximum at x=0. The relative maximum occurs at (0, 15).

Also, because the function changed direction from decreasing to increasing there will be a relative minimum at x=4. The relative minimum occurs at

(4, -17).
{"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://www.enotes.com/homework-help/f-x-x-3-6x-2-15-find-critical-numbers-open-529660", "fetch_time": 1490519818000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2017-13/segments/1490218189198.71/warc/CC-MAIN-20170322212949-00574-ip-10-233-31-227.ec2.internal.warc.gz", "warc_record_offset": 893437749, "warc_record_length": 12740, "token_count": 412, "char_count": 1472, "score": 3.984375, "int_score": 4, "crawl": "CC-MAIN-2017-13", "snapshot_type": "longest", "language": "en", "language_score": 0.7952640056610107, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00044-of-00064.parquet"}