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Let f(x)=x^(3)+ax^(2)+bx+c be the given ...

Let `f(x)=x^(3)+ax^(2)+bx+c` be the given cubic polynomial and `f(x)=0` be the corresponding cubic equation, where `a, b, c in R.` Now, `f'(x)=3x^(2)+2ax+b`
Let `D=4a^(2)-12b=4(a^(2)-3b)` be the discriminant of the equation `f'(x)=0`.
If `D=4(a^(2)-3b)gt0 and f(x_(1)).f(x_(2))gt0` where `x_(1),x_(2)` are the roots of f(x), then

A

f(x) has all real and distinct roots

B

f(x) has three real roots but one of the roots would be repeated

C

f(x) would have just one real root

D

None of the above

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