December 16th, 2018, 10:31 AM  #1 
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.  Convex Functions
I've ran across two definitions for convex functions. One is taking the second derivative to see if it's positive, and another is some weird definition; $f$ is called convex if: ${\displaystyle \forall x_{1},x_{2}\in X,\forall t\in [0,1]:\qquad f(tx_{1}+(1t)x_{2})\leq tf(x_{1})+(1t)f(x_{2}).} \forall x_{1},x_{2}\in X,\forall t\in [0,1]:\qquad f(tx_{1}+(1t)x_{2})\leq tf(x_{1})+(1t)f(x_{2}).$ I'm just wondering if these are the same or if they are two different ideas. 
December 16th, 2018, 10:43 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,939 Thanks: 2210 
If the function is twice differentiable, testing the second derivative suffices. Otherwise, the "weird" definition (or an equivalent definition) should be used.

December 16th, 2018, 10:46 AM  #3 
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.  
December 16th, 2018, 10:57 AM  #4 
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  
December 16th, 2018, 06:33 PM  #5 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,834 Thanks: 650 Math Focus: Yet to find out.  
December 17th, 2018, 10:35 AM  #6 
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. 
Wow! Thank you! Sweet!


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