Home
Class 12
MATHS
Prove that a^4+b^4+c^4> a b c(a+b+c),w h...

Prove that `a^4+b^4+c^4> a b c(a+b+c),w h e r ea ,b ,c > 0.`

Text Solution

Verified by Experts

`(a^4+b^4+c^4)/(3)gt((a+b+c)/(3))^4`
`=((a+b+c)/(3))((a+b+c)/(3))^3 `
Now,
`(a+b+c)/(3)gt (abc)^(1//3) or ((a+b+c)/(3)^3 gt abc`
`rArr (a^4+b^4+c^4)/(3)gt((a+b+c)/(3))abc`
or `a^4+b^4+c^4 gt abc (a+b+c)`.
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise EXERCISES (Single Correct answer type)|20 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Multiple correct answers type|5 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Concept Application Eexercises 6.3|6 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

Given a matrix A=[a b c b c a c a b],w h e r ea ,b ,c are real positive numbers a b c=1a n dA^T A=I , then find the value of a^3+b^3+c^3dot

Prove that a^4+b^4+c^4>abc(a+b+c). [a,b,c are distinct positive real number]..

If |[1 1 1];[a b c];[ a^3b^3c^3]|=(a-b)(b-c)(c-a)(a+b+c),w h e r ea ,b ,c are different, then the determinant |[1 1 1];[(x-a)^2(x-b)^2(x-c)^2];[(x-b)(x-c) (x-c)(x-a) (x-a)(x-b)| vanishes when a. a+b+c=0 b. x=1/3(a+b+c) c. x=1/2(a+b+c) d. x=a+b+c

Suppose A, B, C are defined as A=a^2b+a b^2-a^2c-a c^2 , B=b^2c+b c^2-a^2b-a b^2 - bc^2, and C=a^2c +'a c^2-b^2' c-b c^2, w h e r ea > b > c >0 and the equation A x^2+B x+C=0 has equal roots, then a ,b ,c are in AdotPdot b. GdotPdot c. HdotPdot d. AdotGdotPdot

Use the factor theorem to find the value of k for which (a+2b),w h e r ea ,b!=0 is a factor of a^4+32 b^4+a ^3 b(k+3)dot

Evaluate int_a^b(dx)/(sqrt(x)),w h e r ea , b > 0.

Prove that a cos A + b cos B + c cos C = 4 R sin A sin B sin C.

Prove that {:[( b+c,a,a), ( b,c+a,b),( c,c,b+a) ]:} = 4abc

If the sum of n terms of an A.P. is given by S_n=a+b n+c n^2, w h e r ea ,b ,c are independent of n ,t h e n (a) a=0 (b) common difference of A.P. must be 2b (c) common difference of A.P. must be 2c (d) first term of A.P. is b+c

Prove that |a ,b+c, a^2,b, c+a, b^2,c, a+b, c^2|=-(a+b+c)xx(a-b)(b-c)(c- a)

Home
Profile