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

This Question is Closed

johnweldon1993
 one year ago
Best ResponseYou've already chosen the best response.0Critical points are found via taking the first derivative of the function...and setting it = to 0 So what is \[\frac{ d }{ dx }x^44x^3\]

johnweldon1993
 one year ago
Best ResponseYou've already chosen the best response.0One step ahead of me...so you will have 2 critical points.. being...?

johnweldon1993
 one year ago
Best ResponseYou've already chosen the best response.0That would be correct...Now do you have to classify if they are a max or a min or an inflection point?

burhan101
 one year ago
Best ResponseYou've already chosen the best response.0How do i find that out?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0plug them into the original function and calculate its value. f(0) =? and f(3)=? compare them to give out the answer

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0ok, between them, which one is bigger? the bigger is local max, the smaller is local min. right?

burhan101
 one year ago
Best ResponseYou've already chosen the best response.0what do you mean which one :S

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0for your problem, so far you have 4 critical points, 4, 0, 3, 4, just plug them into the original function and make conclusion. done.

burhan101
 one year ago
Best ResponseYou've already chosen the best response.0where did 4 and 4 come from ?

oldrin.bataku
 one year ago
Best 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.

oldrin.bataku
 one year ago
Best 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

burhan101
 one year ago
Best ResponseYou've already chosen the best response.0@oldrin.bataku what do i do with these points now?

burhan101
 one year ago
Best ResponseYou've already chosen the best response.0plug them into f(x)?

oldrin.bataku
 one year ago
Best 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)

burhan101
 one year ago
Best ResponseYou've already chosen the best response.0why are we plugging in 3 ?
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