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 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: ${\displaystyle \forall x_{1},x_{2}\in X,\forall t\in [0,1]:\qquad f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).} \forall x_{1},x_{2}\in X,\forall t\in [0,1]:\qquad f(tx_{1}+(1-t)x_{2})\leq tf(x_{1})+(1-t)f(x_{2}).$ 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. Thanks from ProofOfALifetime
December 16th, 2018, 10:46 AM   #3
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Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.
Quote:
 Originally Posted by skipjack If the function is twice differentiable, testing the second derivative suffices. Otherwise, the "weird" definition (or an equivalent definition) should be used.
Thank you! I guess it's not that weird, it's a good definition. It's just unfamiliar to me!

December 16th, 2018, 10:57 AM   #4
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Quote:
 Originally Posted by ProofOfALifetime Thank you! I guess it's not that weird, it's a good definition. It's just unfamiliar to me!
It helps to draw a picture, since the concept is much simpler than the formal definition.

 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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