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Here's a homework question I'm struggling with:

Prove/disprove the next statement:

Let $f,g$ two convex functions, then $h(x)=f(x) \cdot g(x)$ is also convex

So, we know that $h'(x)=f'(x) \cdot g(x) + f(x) \cdot g'(x)$. We also know that $f'(x),g'(x)$ are monotonically increasing because they are convex. If I can show that $h'(x)$ is also monotonically increasin I'm done, but I'm not sure how to do it. Any hints?

Thanks!

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2  
Try to find a counterexample ... :D –  martini May 21 '12 at 8:14
    
That didn't help me too mcuh, sorry –  yotamoo May 21 '12 at 8:25
    
Ok ... $x \mapsto (x-a)^2$ is convex for any $a$, right :) –  martini May 21 '12 at 8:28

1 Answer 1

up vote 2 down vote accepted

Hint: You can write the function defined by $x\mapsto-x^2$ as the product of two very simple linear and hence convex functions.

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So the statement is not even true. I assumed it is. Thanks! –  yotamoo May 21 '12 at 8:37
    
I am sorry but I am new to convex functions. Is the statement "the product of convex functions is convex" true or false ? Could you please clarify and give more details. For example if $f(x)=-x^2$ and $g(x)=-x^2$ then $fg$ is a convex function whereas $f$ and $g$ are not. –  roni Aug 25 at 7:41
1  
@roni It is false. Take the two functions $x\mapsto x$ and $x\mapsto -x$. –  Michael Greinecker Aug 25 at 16:40

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