Constrained minimization problem of a parabola

Question

For which positive value(s) of $x$ the following function attains the lowest value

$f(x) = x^2 + ax +c$ [ where $a ,c > 0$ ]

[note : I know there is no positive $x$ for which $f(x)$ is minimum but I want the value of $x$ for which $f(x)$ is minimum among all $f(x^{*})$ where $x^{*}>0$; I don't want heuristics algorithms to solve this. I want a mathematical way to find $x$ and a proof would be great. Any optimal control theory based approach also would be great.]

Thanks and regards

Answer 1

Well, the vertex of the function occurs at $x=-\frac{a}2<0$ and the parabola opens upward. Thus, the function has no minimum on the positive reals, but the infimum of its function-values over the positive reals is $f(0)=c$.