Lower bound for $-a \operatorname{ln}x +bx$

Question

Let $a,b>0$ constants and $x \in \mathbb{R}^+$. I have shown that $-a \operatorname{ln}x +bx \ge 0$ holds for $x\le 1$ and for $x>1, a\le b$. I don`t know how to get a lower bound for $x>1, a> b$. Does someone have an idea?

Answer 1

It depends what you mean for a lower bound. Suppose $a>b$, then the derivative of the function is zero at $x=a/b>1$. The function is convex, so this is a global minimizer and has value $-a\ln (a/b)+a$. This is the best lower bound in terms of a constant you can get.