Let `f(x)=x^(3)-9x^(2)+24x+c=0` have three real and distinct roots `alpha, beta` and `lambda`.
(i) Find the possible values of c.
(ii) If `[alpha]+[beta]+[lambda]=8`, then find the values of c, where `[*]` represents the greatest integer function.
(ii) If `[alpha]+[beta]+[lambda]=7`, then find the values of c, where `[*]` represents the greatest integer function.s

Options

A. (-20,-16)
B. (-20,-18)
C. (-18,-16)
D. none of these

Answer

2

Explanation

`Let g(x) =x^(3)-9x^(2)+24x=x(x^(2)-9x+24)`
`therefore g(X)=3(x-2)(x-4)`
Singn scheme of g(X)

for three real roots of f(X)=`x^(3)-9x^(2)+24+c=0`
in the interval (-20,-16)
`f(x)=clt0`
`f(1)=1-9+24+c=c+16lt0 for forall c in (-20,-1)`
`f(2)=8-36+48+c=c+20gt0`
`alphain (1,2) or [alpha]=1`
`therefore beta in (3,4)if c in (-18,-16)`
now `f(4)=64-144+96+c=16+clt0 if cin (-18,-16)`
or `betan in (3,4) if c in (-18,-16)`
Now `f(4)=64-144+96+c=16+clt0 forall c in (-20,16)`
or ` gamma(4,5) or [gamma]=4`
Thus `[alpha]+[beta]+[gamma]={{:(1+2+4-20ltclt-18),(1+3+4,-18l,tclt-16):}`
Now if `c in (-20,-18)`
`alphain (1,2) beta (2,3)gamma in (4,5)`
or `[alpha]+[beta]+[gamma]=7`
If `c in (-18,-16),alpha in (1,2) , beta (3,4) gamma in (4,5) then [alpha+[beta]+[gamma]=8`