If $ f(x)$ be a concave function, what can be said about the convexity of $f(x)=1/g(x)$
I thinking don't have convexity for $1/g(x)$ because maybe $g(x)=0$ , $g(x)\leq 0$
But I have no idea for proof, my guess is true? how can Begin the proof?
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$\begingroup$ So a related question. If you know $f''(x) < 0$ for all $x$, what is the sign of $(1/f(x))''$ ?? $\endgroup$– GEdgarCommented Nov 16, 2017 at 16:56
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$\begingroup$ Please consider reading, say, the first three chapters of Boyd & Vandenberghe. This very question is addressed there. $\endgroup$– Michael GrantCommented Nov 16, 2017 at 19:14
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$\begingroup$ That said: when you are trying to disprove convexity, you only need one counterexample. So you need just one concave function whose reciprocal is not convex. That's not hard, don't you think? $\endgroup$– Michael GrantCommented Nov 16, 2017 at 19:15
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$\begingroup$ @Michael Grant thanks, your recommended is very good :-) $\endgroup$– yaodao vangCommented Nov 17, 2017 at 5:08
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$\begingroup$ @Michael Grant , please give me an example, thanks, I try to make a counter example but, I can't:-(( $\endgroup$– yaodao vangCommented Nov 17, 2017 at 5:53
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