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Given-m-gt-0-n-gt-0-m-n-a-Find-the-range-of-a-such-that-m-1-m-n-1-n-gets-its-minimum-iff-m-n-




Question Number 215883 by CrispyXYZ last updated on 20/Jan/25
Given m>0, n>0, m+n=(√a).  Find the range of a such that  “(m+(1/m))(n+(1/n)) gets its minimum iff m=n”.
Givenm>0,n>0,m+n=a.Findtherangeofasuchthat(m+1m)(n+1n)getsitsminimumiffm=n.
Commented by MathematicalUser2357 last updated on 21/Jan/25
“iff”? does it mean ifh.cc?
iff?doesitmeanifh.cc?
Commented by CrispyXYZ last updated on 21/Jan/25
“if and only if”.
ifandonlyif.
Answered by Eze120 last updated on 23/Jan/25
m=n  (m+m^(−1) )(m+m^(−1) )=m^2 +(m^(−1) )^2   2m=(√a)  2m^2 =a  m^2 =(a/2)  (m^(−1) )^2 =(1/m^2 )=(1/(a/2))  (a/2)+(1/(a/2))=(1/2)a+(2/a)  PD:It′s a guess, it′s not accurate.
m=n(m+m1)(m+m1)=m2+(m1)22m=a2m2=am2=a2(m1)2=1m2=1a/2a2+1a/2=(1/2)a+2aPD:Itsaguess,itsnotaccurate.

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