Let \(f_1(x)=x^2+a_1x+b_1\) and \(f_2(x)=x^2+a_2x+b_2\) be polynomials with real coefficients. Prove or disprove that the following are equivalent.
(i) There exist two positive reals \(c_1, c_2\) such that \[ c_1f_1(x)+ c_2 f_2(x) > 0\] for all reals \(x\).
(ii) There is no real \(x\) such that \( f_1(x)\le 0\) and \( f_2(x)\le 0\).
GD Star Rating
loading...
2018-22 Two monic quadratic polynomials,
loading...