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Determine and classify all critical points of each function on the interval 4<x<4
f(x) = x⁴4x³
 10 months ago
 10 months ago
Determine and classify all critical points of each function on the interval 4<x<4 f(x) = x⁴4x³
 10 months ago
 10 months ago

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johnweldon1993Best ResponseYou've already chosen the best response.0
Critical points are found via taking the first derivative of the function...and setting it = to 0 So what is \[\frac{ d }{ dx }x^44x^3\]
 10 months ago

johnweldon1993Best ResponseYou've already chosen the best response.0
One step ahead of me...so you will have 2 critical points.. being...?
 10 months ago

johnweldon1993Best ResponseYou've already chosen the best response.0
That would be correct...Now do you have to classify if they are a max or a min or an inflection point?
 10 months ago

burhan101Best ResponseYou've already chosen the best response.0
How do i find that out?
 10 months ago

Loser66Best ResponseYou've already chosen the best response.0
plug them into the original function and calculate its value. f(0) =? and f(3)=? compare them to give out the answer
 10 months ago

Loser66Best ResponseYou've already chosen the best response.0
ok, between them, which one is bigger? the bigger is local max, the smaller is local min. right?
 10 months ago

burhan101Best ResponseYou've already chosen the best response.0
what do you mean which one :S
 10 months ago

Loser66Best ResponseYou've already chosen the best response.0
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.
 10 months ago

burhan101Best ResponseYou've already chosen the best response.0
where did 4 and 4 come from ?
 10 months ago

oldrin.batakuBest ResponseYou've already chosen the best response.2
$$f(x) = x^44x^3\\f'(x)=4x^312x^2=4x^2(x3)\\4x^2(x3)=0\implies4x^2=0,x3=0$$... so we conclude \(x=0\) and \(x=3\) are our critical points. Both lie in our interval so classify them.
 10 months ago

oldrin.batakuBest ResponseYou've already chosen the best response.2
@burhan101 they're our endpoints, which you'd want to test when finding *absolute* extrema. @Loser66 had a minor misunderstanding
 10 months ago

burhan101Best ResponseYou've already chosen the best response.0
@oldrin.bataku what do i do with these points now?
 10 months ago

burhan101Best ResponseYou've already chosen the best response.0
plug them into f(x)?
 10 months ago

oldrin.batakuBest ResponseYou've already chosen the best response.2
@burhan101 no you want to check the second derivative:$$f''(x)=12x^224x=12x(x2)\\f''(0)=0\\f''(3)=12(3)=36>0$$... 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)
 10 months ago

burhan101Best ResponseYou've already chosen the best response.0
why are we plugging in 3 ?
 10 months ago
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