## Problem of the week

### 2018-22 Two monic quadratic polynomials

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 $$f_2(x)\le 0$$.