1. The problem statement, all variables and given/known data a) Let f(x) = ax^2+ bx + c; ∀x ∈ ℝ; where a(≠ 0), b and c ∈ ℝ are constants, (i) Prove that f(k) =(64+(b^2−4ac)^2)/(64a) ; where k = −(b/2a)+(1/a)+(b^2−4ac)/(8a). (ii) Hence, without using graphs and without using your knowledge on quadratic functions and equations, prove that (∀x ∈ ℝ f(x) > 0) [itex]\Rightarrow[/itex] (a > 0 and b^2 − 4ac< 0). 2. Relevant equations 3. The attempt at a solution i have already proven the first part but i m stuck in the 2nd part that says prove from "hence" . otherwise i can show that there exist M in R such that for all x in R ,M<=f(x) and M=inf(f(x)), and there exist x' in R such that f(x')=M, so i can take an argument that x' to be in real and if this happens (a>0 and b^2-4ac<0) ,M>0, so f(x)<0, but on that approach i m not using any above proven things to get my second answer(hence). if any one can help would be great. sorry for my english.