Can we say that \(\displaystyle\forall{x}\in{\mathbb{{{R}}}},{g{{\left({x}\right)}}}{>}{0},{g{{\left({x}\right)}}}{>}{0}\) or that

Payton Benson

Payton Benson

Answered question

2022-03-16

Can we say that xR,g(x)>0,g(x)>0 or that (xR)g(x)=0.

Answer & Explanation

PietAppeteoth22m

PietAppeteoth22m

Beginner2022-03-17Added 5 answers

Using the "vertex form" of the polynomials, we are taking
f(x)=ax2+bx+c=a(x+b2a)2+(cb24a)>0,
which then requires cb24a>0. [Naturally, this is equivalent to the discriminant condition.]
For  g(x) = f(x)+f(x)+f(x) = ax2+(2a+b)x+(2a+b+c)  ,
then g(x)  =  a·(x + (2a + b)2a)2 + ( [2a+b+c]  (2a + b)24a)
=  a·(x + [ b2a+1 ])2 + ( [2a+b+c]  [a+b+b24a] )
=  a·(x + [ b2a+1 ])2 + ( a+[ cb24a ] )  .
The vertex of the parabola corresponding to g(x) is therefore always one unit "to the left" and a units "above" the vertex for f(x) with a>0, guaranteeing that g(x)>0.
produsint8l5

produsint8l5

Beginner2022-03-18Added 4 answers

We can conclude g(x)>0 everywhere. If f(x)>0 then a>0, and since the coefficient of x2 in g(x) is also a, and the discriminant of g is negative, we must have g(x)>0 for all x.

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