A quadratic ax^{2}+bx+c has its roots in the interval [0,1],

trasdulaive

trasdulaive

Answered question

2022-03-03

A quadratic ax2+bx+c has its roots in the interval [0,1], find the maximum value of (ab)(2ab)a(ab+c)

Answer & Explanation

nghycylluzn0

nghycylluzn0

Beginner2022-03-04Added 4 answers

Since ax2+bx+c=0 has the same roots as x2+bax+ca=0 for a0, this suggests writing instead
(ab)(2ab)a(ab+c)=(1ba)(2ba)1ba+ca
and letting p=ba,q=ca, for which we now seek to maximize
f(p,q)=(1p)(2p)1p+q.
But if r1,r2[0,1] are the roots, we also have
(xr1)(xr2)=x2+px+q,
hence p=(r1+r2),q=r1r2,
after which we find
f(r1r2,r1r2)=2+r11+r2+r21+r1.

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