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If L, m, n be the three positive roots of the equation x3 -ax2 +bx -48 = 0 then the minimum value 1/L + 2/m + 3/n = a) 1 b) 2 c) 3/2 d) 5/2

If L, m, n be the three positive roots of the equation x3 -ax2 +bx -48 = 0 then the minimum value 1/L + 2/m + 3/n =
a) 1
b) 2
c) 3/2
d) 5/2

Grade:11

1 Answers

AKASH GOYAL AskiitiansExpert-IITD
420 Points
13 years ago

Dear Ajay

You have to apply Airthmetic mean(AM)≥Geometric mean(GM)

AM=(1/L + 2/m + 3/n)/3

GM=(1/L*2/m*3/m)1/3=(6/Lmn)1/3

From cubic equation product of roots Lmn=-(-48/1)=48

GM=(6/48)1/3 = (1/8)1/3=1/2

apply AM≥GM

(1/L + 2/m + 3/n)/3 ≥ 1/2

(1/L + 2/m + 3/n) ≥ 3/2

hence min value is 3/2  option c

 

All the best.                                                           

AKASH GOYAL

AskiitiansExpert-IITD

 

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