Question

If x ^ 3 - p * x ^ 2 + qx - 27 = 0 has three positive roots where p, q > 0 and p+k+l=4/+ 10 where k 1 I in R , then the greatest value of k is
(C) 6
(D) 7
(A) 1
(B) 5​

Answer 1

Step-by-step explanation:

is B)

Given x

3

+px

2

+qx−q=0

Roots of equation: α,β,λ for HP

General cubic equation ax

3

+bx

2

+cx+d

α+β+λ=(−1)

1

a

b

=(−1)

1

p

=−p

αβ+βλ+αλ=(−1)

2

a

c

=q

αβλ=(−1)

3

1

−q

=q

AM ≥ GM

α

1

+

β

1

+

γ

1

≥3[

αβγ

1

]

1/3

αβγ

γβ+αγ+αβ

≥3[

αβγ

1

]

1/3

q

q

≥3[

αβγ

1

]

1/3

1≥3[

αβγ

1

]

1/3

(1)

α+β+γ≥3(αβγ)

1/3

(2)

αβ+βγ+γα≥3(αβγ)

2/3

(3)

Dividing (2) by (3)

αβ+βγ+γα

α+β+γ

≥

(αβγ)

1/3

1

q

−p

≥

(αβγ)

1/3

1

(4)

Dividing (4) by (1)

q

−p

≥

3

1

q

p

≥−

3

1