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

Explanation

We have `y=f(x)=x^(3)-9x^(2)+24x+c`
`f'(x)=3x^(2)-18x+24=3(x-2)(x-4)`
Sign scheme of `f'(x)` is as follows.

x=4 is the point of local minima and x = 2 is the point of local maxima.
`f(2)=20, f(4)=16`
`If c=0, " then " f(0)=0`
So the graph of `y=f(x)` for `c=0` can be draws as follows.

From the figure, for three real roots of `f(x)=x^(3)-9x^(2)+24x+c=0`, c must lie in the interval `(-20, -16)`.
`:. f(0)=c lt 0`
`f(1)=1-9 +24+c=c+16 lt 0, AA c in (-20, -16)`
`f(2)=8-36+48+c=c+20 gt 0, AA c in (-20, -16)`
`:. alpha in (1,2) rArr [alpha]=1`
`f(3)=27-81+72=18+c`
`rArr f(3) gt 0 " if " c in (-20, -18) " or " f(3) lt 0 " if " c in (-18, -16)`
or ` beta in (2, 3) " if " c in (-18, -16)`
and `beta in (3, 4) " if " c in (-20, -18)`
Now `" " f(4)=64-144+96+c=16 +c lt0, AA c in (-20, -16)`
`f(5) = 125 - 225 +120 +c = c+ 20 gt 0, AA c in (-20, -16)`
` :. lambda in (4, 5) rArr [lambda] = 4`
Thus `" " [alpha]+[beta]+[lambda]= {{:(1+3+4", " -20 lt c lt -18),(1+2+4", " -18 lt c lt -16):}`