If i am given with the equation $$x^2- \alpha_1 x + \alpha_2 \geq 0$$ where $\alpha_1 , \alpha_2 > 0$ and $x > 0$. Is it possible to find out an upper bound on the value of $x$?
I tried to proceed with the help of completing the square but then i am getting an lower bound.
It is impossible to find an upper bound because it does not exist. We can conclude this by observing the behaviour of the polynomial as $x→∞$, which again goes to $∞$ which is greater than zero.
If this solution seems too intuitive we can see from a general graph of a quadratic polynomial ($y=f(x)$)with positive leading coefficient that to get a positive value of $y$, there are two cases:-
if the discriminant is positive then $y>0 Ɐx∈R$.
if it is negative than $y \ge 0$ for all values of $x\ge a$ where a is greater zero of the polynomial(if there is only one zero then a is that root.)