Solve f(x)=x^3-6x^2+15: Help Needed

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Homework Help Overview

The discussion revolves around the function f(x) = x^3 - 6x^2 + 15, focusing on finding intervals of increase/decrease, local maxima/minima, intervals of concavity, and inflection points. Participants are attempting to analyze the function's behavior through its first and second derivatives.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants are calculating the first derivative to find critical points and determining intervals of increase and decrease based on the sign of the derivative. They are also exploring the second derivative to assess concavity and inflection points. Some participants question the correctness of the derivative calculations and the implications of the sign of the derivative.

Discussion Status

There is an ongoing examination of the derivative calculations, with some participants suggesting corrections and others acknowledging potential mistakes. The discussion includes varied interpretations of the function's behavior in different intervals, particularly concerning positivity and negativity.

Contextual Notes

Participants are working under the constraints of homework rules, which require them to explore the problem without providing complete solutions. There is a noted confusion regarding the positivity of the function in certain intervals, particularly for large negative values of x.

hectorubie
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Homework Statement


hey guys. please help me out i tried the problem but for some reason i feel like its wrong. thank you in advance.

Homework Equations


given f(x)=x^3-6x^2+15
a)find the intervals of inc/dec
b)find the local max/min if any
c)find intervals of concavity
d)find the inflection points if any
e)use the information to sketch the graph

The Attempt at a Solution



f'(x)=3x^2-32x
=x(3x-32)
x=0 x=32/3 (critical points)

from -infinity to 0 the function is positive
from 0 to the function is negative
from 32/3 to infinity the function is positive

f(0) = 15 (0,15) = local min
f(32/5) = 35.7 = local max

f''(x)=6x-32
=6x-32=0
=16/3

from -infinity to 16/3 the function is concave down
from 16/3 to infinity the function is concave up

point of inflection (16/3, 15)

cant show the graph here
 
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hectorubie said:

Homework Statement


hey guys. please help me out i tried the problem but for some reason i feel like its wrong. thank you in advance.

Homework Equations


given f(x)=x^3-6x^2+15
a)find the intervals of inc/dec
b)find the local max/min if any
c)find intervals of concavity
d)find the inflection points if any
e)use the information to sketch the graph

The Attempt at a Solution



f'(x)=3x^2-32x
=x(3x-32)
x=0 x=32/3 (critical points)

from -infinity to 0 the function is positive
from 0 to the function is negative
from 32/3 to infinity the function is positive

f(0) = 15 (0,15) = local min
f(32/5) = 35.7 = local max

f''(x)=6x-32
=6x-32=0
=16/3

from -infinity to 16/3 the function is concave down
from 16/3 to infinity the function is concave up

point of inflection (16/3, 15)

cant show the graph here

Your derivative is wrong. Try and fix that and try again. And whether the derivative is positive or negative tells you whether the function is increasing or decreasing, yes?
 
Last edited:


Your derivative is wrong. Try and fix that and try again. And whether the derivative is positive or negative tells you whether the function is increasing or decreasing, yes?

the first derivative is 3x^2-32x
i factored it to x(3x-32). is that wrong?
 


hectorubie said:

Homework Statement


hey guys. please help me out i tried the problem but for some reason i feel like its wrong. thank you in advance.


Homework Equations


given f(x)=x^3-6x^2+15
a)find the intervals of inc/dec
b)find the local max/min if any
c)find intervals of concavity
d)find the inflection points if any
e)use the information to sketch the graph


The Attempt at a Solution



f'(x)=3x^2-32x
=x(3x-32)
x=0 x=32/3 (critical points)

from -infinity to 0 the function is positive
from 0 to the function is negative
from 32/3 to infinity the function is positive

f(0) = 15 (0,15) = local min
f(32/5) = 35.7 = local max

f''(x)=6x-32
=6x-32=0
=16/3

from -infinity to 16/3 the function is concave down
from 16/3 to infinity the function is concave up

point of inflection (16/3, 15)

cant show the graph here

The function f(x) =x^3-6x^2+15 is NOT positive on (-∞,0): when x is large and negative the x^3 term is swamps all the others and is < 0, so f(x) < 0.
 


Ray Vickson said:
The function f(x) =x^3-6x^2+15 is NOT positive on (-∞,0): when x is large and negative the x^3 term is swamps all the others and is < 0, so f(x) < 0.

i see where i did wrong there. thanks
 


i see where you say the deriv was wrong...
the derive is 3x^2-12x :smile: sorry i didnt notice that i did that
 

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