Home
Class 12
MATHS
If a,cgt0 such that a^3+b^3=2 , then sh...

If a,`cgt0` such that ` a^3+b^3=2` , then show that `a+ble 2`.

Text Solution

Verified by Experts

We have
`((a^(3))^(1//3) + (b^(3))^(1//3))/(2) le (a^(3 + b^(3))/(2))^(1//3)`
`implies (a + b)/(2) le 1`
`implies a + b le 2`
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise 6.1|8 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise 6.2|6 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Comprehension|2 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,quad >0 such that a^(3)+b^(3)=2, then show that a+b<=2.

If a gt 0,bgt 0,cgt0 and 2a +b+3c=1 , then

Statement-1: If a, b are positive real numbers such that a^(3)+b^(3)=16 , then a+ble4. Statement-2: If a, b are positive real numbers and ngt1 , then (a^(n)+b^(n))/(2)ge((a+b)/(2))^(n)

If (a + 1/a )^2 = 3 and a ne 0 , then show that a^3 + 1/a^3 = 0

If (a + 1/a )^2 = 3 and a ne 0 , then show that a^3 + 1/a^3 = 0

If a, b, c and d are in proportion, then show that (a^(3)+3ab^(2))/(3a^(2)b+b^(3))=(c^(3)+3cd^(2))/(3c^(2)d+d^(3))

If the equation a x^2+b x+c=0(a >0) has two real roots alphaa n dbeta such that alphalt-2 and betagt2, then which of the following statements is/are true? (a)a-|b|+c<0 (b)clt0,b^2-4a cgt0 (c) 4a-2|b|+c<0 (d) 9a-3|b|+c<0

If the acute angle between the lines ax^2+2hxy +by^2=0 is 60^@ then show that (a+3b)(3a+b)= 4h^2

For a,b,cgt0 , and a+b+c = 2 , and the maximum value of a^2b^2c^2 is k , then [k]=? (where [.] is G.I.F.)

CENGAGE-INEQUALITIES INVOLVING MEANS -Examples
  1. Prove that (sec^4alpha)/(tan^2beta)+(sec^4beta)/(tan^2alpha)ge8. If ea...

    Text Solution

    |

  2. Prove that [(x^2+y^2+z^2)/(x+y+z)]^(x+y+z)gtx^xy^yz^zgt[(x+y+z)/(3)]...

    Text Solution

    |

  3. Prove that 1^1xx2^2xx3^3xx...xn^nlt ((2n+1)/(3))^((n(+1))/(2)

    Text Solution

    |

  4. Find the greatest value of x^2y^3, w h e r exa n dy lie in the first q...

    Text Solution

    |

  5. Find the maximum value of (7-x)^4(2+x)^5w h e nx lies between -2a n d7...

    Text Solution

    |

  6. Find the maximum value of xyz when (x)/(1)+(y^2)/(4)+(z^3)/(27)=1, whe...

    Text Solution

    |

  7. If a,cgt0 such that a^3+b^3=2 , then show that a+ble 2.

    Text Solution

    |

  8. If m >1,n in N show that 1m+2m+2^(2m)+2^(3m)++2^(n m-m)> n^(i-m)(2^n-...

    Text Solution

    |

  9. Prove that n in acute angled triangle ABC , sec A+sec B +sec Cge 6.

    Text Solution

    |

  10. Prove that (b^2+c^2)/(b+c)+(c^2+a^2)/(c+a)+(a^2+b^2)/(a+b)gt a+b+c , w...

    Text Solution

    |

  11. Prove that (a^8+b^8+c^8)/(a^3b^3c^3)>1/a+1/b+1/c

    Text Solution

    |

  12. If a ,b ,a n dc are positive and a+b+c=6, show that (a+1//b)2+(b+1//c)...

    Text Solution

    |

  13. Let x1, x2, ,xn be positive real numbers and we define S=x1+x2++xndot...

    Text Solution

    |

  14. If 2x^3 +ax^2+ bx+ 4=0 (a and b are positive real numbers) has 3 real ...

    Text Solution

    |

  15. Calculate the greatest and least values of the function f(x)=(x^4)/(x...

    Text Solution

    |

  16. If a,b,c are three distinct positive real numbers in G.P., than prove ...

    Text Solution

    |

  17. If x and y are real numbers such that 16^(x^(2)+y)+16^(x+y^2)=1, then ...

    Text Solution

    |

  18. In how many parts an integer N>=5 should be dissected so that the pro...

    Text Solution

    |

  19. If x+y+z=1a n dx ,y ,z are positive, then show that (x+1/x)^2+(y+1/y)^...

    Text Solution

    |

  20. If a, b,c are three positive real numbers , then find minimum value of...

    Text Solution

    |