IIT JEE , AIEEE Forum - Mathematics , Algebra

karan_33

if a1 a2 a3 a4........ an are in hp then

a1/a2+a3+...an , a2/a1+a3+.an,...an/a1+a2+....+an-1 are in

a1 meanns a base 1 similar for all cases

find min value of x^4+1/x^2

computer001

they 2 will be in H>P
consider as t1,t2...tn
t1=a1/(a2+a3...an)
so 1/t1 = (a2+a3+...an)/a1
so 1/t1 + 1 = (a1+a2+a3....an)/a1
similarly
1/t2 +1 =(a1+a2+a3....an)/a2
1/t3 + 1=(a1+a2+a3....an)/a3

1/t3 +1 + 1/t1 +1= (a1+a2+a3....an)/a3 +(a1+a2+a3....an)/a1
                        =(a1+a2...an){1/a1+1/a3}
                         =(a1+a2...an){2/a2} as a1 a2 a3 are in hp
                        =2{1/t2+1}
so 1/t3 +1+ 1/t1 +1=2/t2 + 2
so 1/t3 + 1/t1= 2/t2
hence t1,t2,t3 re in hp
similarly u can account for rest of the terms

nadeemoidu

Let $f(x) = x^4 + \frac{1}{x^2}$
find f '(x) and find the minima. It is obtained at $x={2^{-\frac{1}{6}}}$

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