f(x)=x3-9 x2+24 x+c=0 has three real and distinct roots, α, β, and γ If [α]+[β]+[γ]=7, then the values of c, where [.] represents the greatest integer function, are

Question

f(x)=x3-9 x2+24 x+c=0 has three real and distinct roots, α, β, and &gamma; If [α]+[β]+[&gamma;]=7, then the values of c, where [.] represents the greatest integer function, are (A) (-20, - 16) (B) (-20, -18) (C) (-18, -16) (D) none of these

Tardigrade Admin · Accepted Answer

Let g(x)=x3-9 x2+24 x=x(x2-9 x+24) ⇒ g'(x)=3(x-2)(x-4) Sign scheme of g(x)

For three real roots of f ( x )= x 3-9 x 2+24 x + c =0, c must lie in the interval (-20,-16) f (0)= c <0 f (1)=1-9+24+ c = c +16<0 for ∀ x ∈(-20,-16) f (2)=8-36+48+ c = c +20>0 α ∈(1,2) f (3)=27-81+72+ c =18+ c ⇒ f (3)<0 if c ∈(-20,-18) or f (3)>0 if c ∈(-18,-16) and β ∈(3,4) if c ∈(-18,-16) Now f(4)=64-144+96+c =16+c< 0 ∀ c ∈(-20,-16) f(5)=125-225+120+c =c+20>0 ∀ c ∈(-20,-16) ⇒ γ ∈(4,5) ⇒ [γ]=4 Thus, [α]+[β]+[γ]= begincases1+2+4, -20 < c < -18 1+3+4, -18< c < -16 endcases Now if c ∈(-20,-18) α ∈(1,2), β ∈(2,3), γ ∈(4,5) ⇒ [α]+[β]+[γ]=7 if c ∈(-18,-16), α ∈(1,2), β ∈(3,4), γ ∈(4,5) ⇒ [α]+[β]+[γ]=8

Q. f(x)=x3−9x2+24x+c=0 has three real and distinct roots, α,β, and γ If [α]+[β]+[γ]=7, then the values of c, where [.] represents the greatest integer function, are

(-20, - 16)

(-20, -18)

(-18, -16)

none of these

Q. $f (x) = x^{3} - 9 x^{2} + 24 x + c = 0$ has three real and distinct roots, $α, β$ , and $γ$
If $[α] + [β] + [γ] = 7$ , then the values of $c$ , where [.] represents the greatest integer function, are