For me, possibly the most out-of-nowhere definition of the first semester of Calculus was the following definition of a convex function and its equivalents.
Function $f$ is convex on the interval $J$ if for $\forall x,y\in J$ and $\forall\lambda\in (0,1)$ is $$f(\lambda x + (1-\lambda )y)\leq\lambda f(x) + (1-\lambda )f(y)\tag1$$
Equivalently if $\forall u,v,w\in J:u<v<w$ $$f(v)(w-u)\leq f(w)(v-u)+f(u)(w-v)\tag2$$
I'm looking for an intuition, or visual representation of what these three definitions "actually" mean.
(3), being very similar to the definition of a derivative,
is the only one that makes sense to me, that is: a function is convex if the slope between points $(u,f(u))$ and $(v,f(v))$ is lesser than the slope between $(v,f(v))$ and $(w,f(w))$.
(2) seems to look at areas of rectangles, however, that is about everything I could say about it.
(1) Got it! $f(\lambda x + (1-\lambda )y)$ is the functional value of a point between $x$ and $y$ and $\lambda f(x) + (1-\lambda )f(y)$ is a point between $f(x)$ and $f(y)$ on a slope between the two points, thus represented by the fact that the slope is never below the functional value.
I can now see that it represents the fact that the slope between $x$ and $y$ is always above the function, I don't see, however, how $\lambda f(x) + (1-\lambda )f(y)$ is a point on the slope.
Thanks for any help!