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If 2 a+3 b+6 c=0(a, b, c \in R), then the quadratic equation a x^2+b x+c=0 has

 

Option: 1

 at least one root in (0,1)
 


Option: 2

at least one root in [2,3]
 


Option: 3

 at least one root in [4,5]
 


Option: 4

none of these


Answers (1)

best_answer

 Let us consider \mathrm{f(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x }
\mathrm{\therefore \quad f(0)=0\, and f(1)=\frac{a}{3}+\frac{b}{2}+c=\frac{2 a+3 b+6 c}{6}=0(given) }
As  \mathrm{f(0)=f(1)=0\, and\, f(x) } is continuous and differentiable also in [0,1].
\mathrm{\therefore \quad\, \, By\, \, Rolle's\, \, theorem,\, \, f^{\prime}(x)=0 }
 \mathrm{\Rightarrow a x^2+b x+c=0 } has at least one root in the interval (0,1).

Posted by

himanshu.meshram

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