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Consider a function $f$ defined over an interval $[a,b]$.

We say that $f$ is convex over $[a,b]$ if, for every pair of points $d,p$ in $[a,b]$ such that $d<p$, the slope of $f$ at $p$ is greater than or equal to the slope of $f$ at $d$.

So, as we sweep across $[a,b]$ from left to right, the slope of $f$ never decreases (and in the case of strict convexity, the slope is always increasing as we sweep across the interval).

And we say that $f$ is concave over $[a,b]$ if the opposite is true, i.e, if, as we sweep across $[a,b]$ from left to right, the slope of $f$ never increases (and in the case of strict concavity always decreases).

Is my understanding correct?

Mahdi Rkioui
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    ConVex : the profile is like a V ($\cup$) ; ConcAve : the profile is like a A ($\cap$) ; your intuition is correct. – Lourrran May 08 '23 at 21:58
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    That is a good intuition of convexity, with the caveat that the slope may not always be defined. For example, $|x|$ is convex on $\mathbb R$ but its slope is undefined at $x=0$. – dxiv May 08 '23 at 22:19
  • @dxiv I haven't studied continuity (or discontinuity) yet. Will I confuse myself if I try to think about special cases like that right now? – Mahdi Rkioui May 08 '23 at 22:25
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    @MahdiRkioui Slope is about differentiability, not continuity. A convex function must be continuous, and $|x|$ is continuous at $0$, but it is not differentiable at $0$ and therefore the slope is undefined there. The intuition is fine, but you have to keep in mind it's just that, an intuition, not a substitute for the formal definition of convexity. – dxiv May 08 '23 at 22:31
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    @dxiv Ok, I think I understand; the formal definition takes into account these special cases while this intuition doesn't. I assume this is why concavity and convexity isn't defined with derivatives. Thanks. – Mahdi Rkioui May 08 '23 at 22:40

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Yes, you are correct and your conclusions about the slope essentially give you an intuition for relating this to second derivative, as it is the derivative of a slope.

Sgg8
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