Let $\;x\in (0,+\infty)\;$ and consider $\;a,c \gt 0\;,b \in \mathbb R\;$ such that:
$\;ax^2\pm 2bx+c \ge 0\;\;\forall x \gt 0\;$. Prove $\;b^2 \le ac\;$
My professor used the following argument:
$\;ax^2 \pm 2bx+c \ge 0 \Rightarrow \frac{ax}{2} + \frac{c}{2x} \ge \mp b\;$ and then by minimization with respect to $\;x\;$, choosing $\;x=(\frac{a}{c})^{1/2}\;$ it follows that $\; \pm b \le (ac)^{1/2}\;$
I 'm having a really hard time keeping up with the above argument. Why is it valid? How did he chose that certain $\;x\;$ and finally how did he deduce to the last inequality?
I proved the same inequality but in a different way:
I wrote $\;D=(\pm2b)^2-4ac=4b^2-4ac=4(b^2-ac)\;$. So if $\;ax^2\pm 2bx+c \ge 0\;\;\forall x \gt 0\;$ then $\;D \le 0\;$ and hence $\;b^2 \le ac\;$
However I want to understand the way my professor used because it pops up quite often during the class.
I would really apreciate if somebody could help me through this.
Thanks in advance!
What your professor did was to search for a number $x_0$ that minimizes the function $f:(0,+\infty)\to \mathbb{R}:x\mapsto \frac{ax}{2}+\frac{c}{2x}$. That such a number exists can be proven by noticing that $$\lim_{x\to +\infty}f(x)=+\infty=\lim_{x\to 0}f(x)$$ and using the fact that $f$, as a continuous function, must reach its lower bound on any compact interval. It is well-known that such a point $x_0$ must be such that $f'(x_0)=0$; so you know that $$\frac{a}{2}-\frac{c}{2x_0^2}=0,$$ which immediately gives you $x_0=\sqrt{\frac{c}{a}}$. Now the fact that $|b|$ is a lower bound for $f$ tells you that in particular $$|b|\leq f(x_0)=\frac{ax_0}{2}+\frac{c}{2x_0}=\frac{a\sqrt{c}}{2\sqrt{a}}+\frac{c\sqrt{a}}{2\sqrt{c}}=\sqrt{ac},$$which is equivalent to $b^2\leq ac$.
Note that in some sense this proof is a little bit "overkill", in the sense that you don't actually need to know that $x_0$ is the minimum point of $f$ for the inequality to hold; but it does give you a way to find what $x_0$ to choose, and it also gives you the best possible bound on $|b|$ that you can extract from the information. For example, if you had tried $x_0=1$, you would have gotten the weaker inequality $|b|\leq\frac{a+c}{2}$.