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If the cubic function f(x)=x^(3)+px+q ha...

If the cubic function `f(x)=x^(3)+px+q` has 3 distinct real roots, then prove that `4p^(3)+27q^(2)lt0`.

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`f(x) =x^(3) =px +q. f(x) =3x^(2)+p`
`:.` f(x) must have one maximum `lt 0` and one minimum `lt 0.f'(x) =0`
`rArr " "x=+- sqrt((-p)/(3)) , p lt0`
f is maximum at `x= - sqrt((-p)/(3)`
`f(-sqrt((-p)/(3))) f (sqrt((-p)/(3))) lt0`
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Statement -1 If the equation (4p-3)x^(2)+(4q-3)x+r=0 is satisfied by x=a,x=b nad x=c (where a,b,c are distinct) then p=q=3/4 and r=0 Statement -2 If the quadratic equation ax^(2)+bx+c=0 has three distinct roots, then a, b and c are must be zero.

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