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120-x-1-x-3-x-5-max-




Question Number 206987 by hardmath last updated on 02/May/24
((120)/(∣x−1∣ + ∣x−3∣ + ∣x−5∣))   ⇒   max = ?
120x1+x3+x5max=?
Answered by Frix last updated on 02/May/24
((120)/(f(x))) is max ⇒ f(x) is min  4≤∣x−1∣+∣x−3∣+∣x−5∣  ⇒ Answer is ((120)/4)=30
120f(x)ismaxf(x)ismin4⩽∣x1+x3+x5Answeris1204=30
Commented by hardmath last updated on 02/May/24
thank you very much dear professor
thankyouverymuchdearprofessor
Answered by A5T last updated on 03/May/24
∣x−3∣=∣3−x∣;∣x−5∣=∣5−x∣; ∣a∣+∣b∣≥∣a+b∣  ∣x−1∣+∣3−x∣≥∣x−1+3−x∣=2...(i)  Similarly, ∣x−1∣+∣5−x∣≥4...(ii)  ∣x−3∣+∣5−x∣≥2...(iii)  (i)+(ii)+(iii)⇒∣x−1∣+∣x−3∣+∣x−5∣≥4  (Equality when x=3)  ⇒((120)/(∣x−1∣+∣x−3∣+∣x−5∣))≤((120)/4)=30.
x3∣=∣3x;x5∣=∣5x;a+b∣⩾∣a+bx1+3x∣⩾∣x1+3x∣=2(i)Similarly,x1+5x∣⩾4(ii)x3+5x∣⩾2(iii)(i)+(ii)+(iii)⇒∣x1+x3+x5∣⩾4(Equalitywhenx=3)120x1+x3+x51204=30.
Commented by hardmath last updated on 03/May/24
thank you so much dear professor
thankyousomuchdearprofessor

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