Critical Points of f(x): -2x^3 + 39x^2 - 180x + 1

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Discussion Overview

The discussion revolves around finding the critical points of the function f(x) = -2x^3 + 39x^2 - 180x + 1, including determining where the function is increasing and decreasing. Participants are also exploring the implications of the first and second derivatives in this context.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant states they found critical points at -2 and 15 after solving the first derivative -6x^2 + 78x - 180 = 0.
  • Another participant suggests that the critical points should be evaluated using the second derivative to determine their nature.
  • A participant claims there was a mistake in solving the quadratic, asserting that the roots are 3 and 15.
  • Another participant provides a different solution, stating the roots are 3 and 10.
  • One participant mentions having computed the second derivative as -12x + 78 and evaluated it at the critical points to classify them as a local minimum and maximum.

Areas of Agreement / Disagreement

There is disagreement regarding the correct critical points, with participants proposing different values (3, 10, -2, and 15) based on their calculations. The discussion remains unresolved as participants have not reached a consensus on the correct critical points.

Contextual Notes

There are unresolved mathematical steps regarding the solving of the quadratic equation, and participants have not clarified the conditions under which their calculations were made.

smith5029
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hi guys I'm stuck on this, i keep thinking i have the right answeres but i can't get them.
the function is f(x): -2x^3 + 39x^2 - 180x + 1
i need to list all of the critical points, and hten indicate where it is increasing and where it is decreasing.

I set the derivative -6x^2 + 78x - 180 = 0 and then solved for x. this is correct right. by doing so i ended up with critical points at -2 and 15. Then made a number line and figured that the slope before -2 was neg, b/w -2 and 15 it was pos, and after 15 it was neg. so my interval of increasing was (-2,15) and the interval of decreasing was (-inf, -2) U (15,inf) where am i going wrong - PLEASE HELP ME
 
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Did u compute the second derivative,too...?To find the nature of those 2 points,u must evaluate the second derivative in those points.

Daniel.
 
You made a mistake in solving the quadratic. The roots are 3 and 15.
 
Yes i determined the second derivative as -12x + 78. Then using the second derivative rule plugged in 15, and -2 telling me that -2 was a local min and 15 was a local max. DO you know if 15, -2 are the only critical points in my problem?
 
i don't see how u got 3 and 15 ??
 
-6x^2 + 78x - 180 = 0

x^2 -13x + 30 = 0
x=(13+/-sqrt(169-4*1*30))/2
x=(13+/-7)/2

x=20/2 = 10
x=6/2= 3
 

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