anonymous
  • anonymous
Determine and classify all critical points of each function on the interval -4
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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johnweldon1993
  • johnweldon1993
Critical points are found via taking the first derivative of the function...and setting it = to 0 So what is \[\frac{ d }{ dx }x^4-4x^3\]
anonymous
  • anonymous
0=4x²(x-3)
johnweldon1993
  • johnweldon1993
One step ahead of me...so you will have 2 critical points.. being...?

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anonymous
  • anonymous
0 and 3 ?
johnweldon1993
  • johnweldon1993
That would be correct...Now do you have to classify if they are a max or a min or an inflection point?
anonymous
  • anonymous
How do i find that out?
Loser66
  • Loser66
plug them into the original function and calculate its value. f(0) =? and f(3)=? compare them to give out the answer
anonymous
  • anonymous
(3, -27)
Loser66
  • Loser66
ok, between them, which one is bigger? the bigger is local max, the smaller is local min. right?
anonymous
  • anonymous
yes
anonymous
  • anonymous
what do you mean which one :S
anonymous
  • anonymous
like between (x,y)
Loser66
  • Loser66
for your problem, so far you have 4 critical points, -4, 0, 3, 4, just plug them into the original function and make conclusion. done.
anonymous
  • anonymous
where did -4 and 4 come from ?
anonymous
  • anonymous
$$f(x) = x^4-4x^3\\f'(x)=4x^3-12x^2=4x^2(x-3)\\4x^2(x-3)=0\implies4x^2=0,x-3=0$$... so we conclude \(x=0\) and \(x=3\) are our critical points. Both lie in our interval so classify them.
anonymous
  • anonymous
@burhan101 they're our endpoints, which you'd want to test when finding *absolute* extrema. @Loser66 had a minor misunderstanding
anonymous
  • anonymous
@oldrin.bataku what do i do with these points now?
anonymous
  • anonymous
plug them into f(x)?
anonymous
  • anonymous
@burhan101 no you want to check the second derivative:$$f''(x)=12x^2-24x=12x(x-2)\\f''(0)=0\\f''(3)=12(3)=36>0$$... so our derivative is increasing near \(x=3\) meaning \(x=3\) is a relative minimum; since our derivative is neither increasing nor decreasing near \(x=0\) we find it's neither! (hence the even multiplicity)
anonymous
  • anonymous
why are we plugging in 3 ?

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