Home
Class 12
MATHS
If a + b =1, a gt 0,b gt 0, prove that (...

If a + b =1, a `gt` 0,b `gt` 0, prove that `(a + (1)/(a))^(2) + (b + (1)/(b))^(2) ge (25)/(2)`

Text Solution

Verified by Experts

We know that A.M. of mth power `gt` mth power of A.M.
` therefore (((a+1)/(a))^2+(b+(1)/(b))^2)/(2)gt [((a+(1)/(b))^2+(b+(1)/(b)))/(2)]^2," here "m=2`
or ` (a+(1)/(a))^2+(b+(1)/(b))^2gt (1)/(2)[(a+b)+((1)/(a)+(1)/(b))]^2`
Also, `(a^-1+b^-1)/(2)gt ((a+b)/(2))^2`
or ` (1)/(2)((1)/(a)+(1)/(b))gt (2)/(a+b)`
or ` (1)/(a)+(1)/(b)gt (4)/(a+b)`
or ` (1)/(a)+(1)/(b)gt4`.
Hence, from (1)
From (i), ` (a+(1)/(a))^2+(b+(1)/(c))^2gt (1)/(2)(1+4)^2=(25)/(2)`
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise (Single)|20 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise (Multiple)|5 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise 6.3|6 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE|Exercise Solved Examples And Exercises|324 Videos

Similar Questions

Explore conceptually related problems

If a+b=1,a>0, prove that (a+(1)/(a))^(2)+(b+(1)/(b))^(2)>=(25)/(2)

If a+b=1, and a,b>0, then prove that (a+(1)/(a))^(2)+(b+(1)/(b))^(2)>=(25)/(2)

If a,b,c are real numbers such that 0 lt a lt 1, 0 lt b gt 1, 0 lt c lt 1, a + b + c = 2 , then prove that (a)/(1 - a) (b)/(1 - b) (c )/(1 - c) ge 8 .

If a,b,c and d are any four consecutive coefficients in the expansion of (1 + x)^(n) , then prove that (i) (a) /(a+ b) + (c)/(b+c) = (2b)/(b+c) (ii) ((b)/(b+c))^(2) gt (ac)/((a + b)(c + d)), "if " x gt 0 .

If a+b+3c=1 and a gt 0, b gt 0, c gt 0 , then the greratest value of a^(2)b^(2)c^(2) is

If a>0,b>0,c>0 prove that a/(b+c)+b/(c+a)+c/(a+b)ge3/2 .

If agt0, b gt0 than lim_(nrarroo) ((a-1+b^((1)/(n)))/(a))^(n)=