One more

Suppose a and b are real numbers such that the roots of the cubic equation ax3-x2+bx-1 = 0 are all positive real numbers. Prove that :

(i) 0 < 3ab ≤1
(ii) b≥√3

P.S. I can't believe I missed this ... this one is easy ...

3 Answers

62
Lokesh Verma ·

Try to use that the derivative of this equation has 2 +ve real roots..

that will give all the things that u want :)

one is derivative +ve

product of roots positive

and sum of roots is +ve..

i think this should give the solution :)

62
Lokesh Verma ·

one thing i missed thoug..

a has to be +ve bcos f(0) is -ve..

so limit to -infinity is -infinity.. other wise there will be a -ve root between -infinity and 0!!

1
varun ·

I did until sum of roots = product of roots.

After that if you use A.M. ≥ G.M. can get the answer ... I forgot about A.M ≥ G.M. in test -.-

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