Thread:

1. Exponential function

How to complete the following table?

And also:
a) f(x) = 2x^3 + 3x^2 - 12x + 6 has critical points at x = -2 and x = 1. Use the second derivative test to determine whether these are local maximum or local minimum values.

b)
how to Calculate f(-2) and f(1) . Briefly explain the appropriateness of these two values
Attached Thumbnails

2. Originally Posted by Snowboarder
How to complete the following table?

And also:
a) f(x) = 2x^3 + 3x^2 - 12x + 6 has critical points at x = -2 and x = 1. Use the second derivative test to determine whether these are local maximum or local minimum values.

b) how to Calculate f(-2) and f(1) . Briefly explain the appropriateness of these two values
$f(x) = 2x^3 + 3x^2 - 12x + 6$

taking the derivative we get

$f'(x)=6x^2+6x-12=6(x^2+x-2)=6(x+2)(x-1)$

So our critical points are -2 and 1

now taking the 2nd derivative we get

$f''(x)=12x+6$

Now we check the above critical points

$f''(-2)=12(-2)+6=-18$ so this is a max

$f''(1)=12(1)+6=18$ so this is a min.

Good luck.
{"extraction_info": {"found_math": true, "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": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.7780370116233826, "perplexity_flag": "middle"}