Home
Class 12
MATHS
Equation x^4+ax^3+bx^2+cx+1=0 has real r...

Equation `x^4+ax^3+bx^2+cx+1=0` has real roots (a,b,c are non-negative).
Minimum non-negative real value of a is

A

10

B

9

C

6

D

4

Text Solution

Verified by Experts

The correct Answer is:
D
Since, a,b,c `ge` 0 roots must be negative. Let the roots be
`x_(1), x_(2),x_(3),x_(4) (lt 0)`. Then.
`x_(1) + x_(2) + x_(3) + x_(4) = -a`
`x_(1)x_(2) + x_(1)x_(3) + x_(1)x_(4) + x_(2)x_(3) + x_(2)x_(4) + x_(3)x_(4) = b`
`x_(1)x_(2)x_(3) + x_(2)x_(3)x_(4) + x_(3)x_(4)x_(1) + x_(3)x_(4) x_(1) + x_(4)x_(1)x_(2) = - c`
`x_(1)x_(2)x_(3)x_(4) = 1`
Now, `((-x_(1)) + (-x_(2)) + (-x_(3)) + (-x_(4)))/(4) ge [(-x_(1))(-x_(2))(-x_(3))(-x_(4))]^(1//4)`
`( :' A.M ge G.M)`
`implies (a)/(4) ge 1`
Hence, the minimum value of a is 4. similarly,
`(x_(1)x_(2) + x_(1) x_(3) + ..... + x_(3)x_(4))/(6) ge [x_(1)^(3) x_(2)^(3) x_(3)^(3) x_(4)^(4)]^(1//4)`
`implies (b)/(6) ge 1`
or `b ge 6`
Hence, the minimum value of is 6. Finally
`(x_(1)x_(2)x_(3) - x_(2)x_(3)x_(4) - x_(3)x_(4)x_(1) - x_(4)x_(1)x_(2))/(4) ge [x_(1)^(3) x_(2)^(3) x_(3)^(3) x_(4)^(3)]^(1//4)`
`implies (c )/(4) ge 1`
or c ge 4`
Hence, the minimum value of c is 4.
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise (Numerical) & JEE Previous Year|11 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise (Multiple)|5 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE|Exercise Solved Examples And Exercises|222 Videos

Similar Questions

Explore conceptually related problems

Equation x^4+ax^3+bx^2+cx+1=0 has real roots (a,b,c are non-negative). Minimum non-negative real value of c is

Equation x^4+ax^3+bx^2+cx+1=0 has real roots (a,b,c are non-negative). Minimum non-negative real value of b is

Let a >0,b >0 and c >0 . Then, both the roots of the equation a x^2+b x+c=0 are a. real and negative b.have negative real parts c. have positive real parts d. None of the above

The equation 4ax^2 + 3bx + 2c = 0 where a, b, c are real and a+b+c = 0 has

If a ,b ,c ,d in R , then the equation (x^2+a x-3b)(x^2-c x+b)(x^2-dx+2b)=0 has a. 6 real roots b. at least 2 real roots c. 4 real roots d. none of these

f(x) is a polynomial function, f: R rarr R, such that f(2x)=f'(x)f''(x). Equation f(x) = x has (A) three real and positive roots (B) three real and negative roots (C) one real root (D) three real roots such that sum of roots is zero

If x ,y ,a n dz are real and different and u=x^2+4y^2+9z^2-6y z-3z x-2x y ,t h e nu is always a. non-negative b. zero c. non-positive d. none of these

If x^(2) +ax - 3x-(a+2) = 0 has real and distinct roots, then minimum value of (a^(2)+1)//(a^(2)+2) is