Home
Class 12
MATHS
If a,b,c are three distinct positive rea...

If a,b,c are three distinct positive real numbers in G.P., than prove that `c^2+2ab gt 3ac`.

Text Solution

Verified by Experts

Since a,b,c are three distinct positive real numbers, so applying `A.M gt G.M` in `c^(2)`, ab,ab we get
`(c^(2) + ab + ab)/(3) gt (c^(2) b^(2) a^(2))^(1//3)`
or `(c^(2) + 2ab)/(3) gt (c^(2) aca^(2))^((1)/(2))` ( `:'` b is G.M of a and c)
or `c^(2) + 2ab gt 3ac`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Example 5|1 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Example 6|1 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Example 3|1 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 three distinct positive real numbers in G.P., then prove that c^2+2a b >3ac

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x 3.

Knowledge Check

  • If a,b,c,d are distinct positive numbers in AP, then

    A
    `adltbc`
    B
    `a+clt b+d`
    C
    `a+d=b+c`
    D
    `(a+1)(d+1)lt(b+1)(cx+1)`

  • 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

    A
    less than 1
    B
    equal to 1
    C
    greater than 1
    D
    any real number

  • If a, b, c are distinct positive rational numbers and they are in A.P., then the roots of the equation ax^(2) +2bx+x=0 are-

    A
    imaginary
    B
    rational and equal
    C
    ral may be rational or irrational
    D
    rational and unequal

    • Similar Questions

    Explore conceptually related problems

    If the parabolas y^2=4a x and y^2=4c(x-b) have a common normal other than the x-axis (a , b , c being distinct positive real numbers), then prove that b/(a-c)> 2.

    If a ,b ,c are three distinct positive real numbers, the number of real and distinct roots of a x^2+2b|x|-c=0 is 0 b. 4 c. 2 d. none of these

    If a ,b , a n dc are distinct positive real numbers such that a+b+c=1, then prove that (1-a)(1-b)(1-c)> 8.

    If a, b and c are three positive real numbers, then show that a^3 + b^3 + c^3 > 3abc .

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

    Home
    Profile