Home
Class 12
MATHS
lf abcd=1 where a,b,c,d are positive re...

lf `abcd=1` where a,b,c,d are positive reals then the minimum value of `a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd` is

Text Solution

Verified by Experts

The correct Answer is:
10
Given,abcd `=1`
We know that , `AM ge GM`
`therefore (a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd)/(10)`
`ge [a^2xxb^2xxc^2xxd^2xx(ab)xx(ac)xx(ad)xx(bc)xx(bd)xx(cd)]^(1//10)`
`=(a^5xxb^5xxc^5xxd^5)^(1//10)`
`=(abcd)^(1//2)`
`=(1)^(1//2)`
`rArr (a^2b^2+c^2+d^2+ab+ac+ad+bc+bd+cd)/(10) ge 1`
`rArr a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd +cd ge 10`.
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Concept Application Eexercises 6.3|6 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Concept Application Eexercises 6.4|4 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Concept Application Eexercises 6.1|8 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE PUBLICATION|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE PUBLICATION|Exercise All Questions|762 Videos

Similar Questions

Explore conceptually related problems

If a,b,c are positive real numbers and 2a+b+3c=1 , then the maximum value of a^(4)b^(2)c^(2) is equal to

If a, b and c are three positive real numbers, show that a^2 + b^2 + c^2 ge ab + bc + ca

Let us factorise the following. a^2 - b^2 + c^2 - d^2 - 2(ac - bd)

If the expression [m x-1+(1//x)] is non-negative for all positive real x , then the minimum value of m must be -1//2 b. 0 c. 1//4 d. 1//2

The real number x when added to its inverse given the minimum value of the sum at x equal to (a) 1 (b) -1 (c) -2 (d) 2

If a,b,c are three distinct positive real number such that a^2+b^2+c^2=1 , then ab+bc+ca=1 is

If a,b,c are distinct positive real numbers and a^(2)+b^(2) +c^(2)=1 then the value of ab+bc+ca is-

If a , ba n dc are the side of a triangle, then the minimum value of (2a)/(b+c-a)+(2b)/(c+a-b)+(2c)/(a+b-c)i s (a) 3 (b) 9 (c) 6 (d) 1

If a, b and c are in H.P., then the value of ((ac + ab - bc)(ab + bc - ac))/((abc)^(2)) is -