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I've always thought that functions with a graph like this:

enter image description here

...are called convex functions. Today, however, while I was going through an economics textbook, this was described as a concave up function. Further, the book also said:

"Quasi-concave functions: these functions have the property that the set of all points for which such a function takes on a value greater than any specific constant is a convex set (i.e., any two points in the set can be joined by a line contained completely within the set"

That's a condition that this function (graphed) seem to be holding. So, is this function convex, concave up or quasi-concave? I understand that something that's concave or convex can also be quasi-concave -- but what is the difference between these different terminologies? Further, it looks like convex and concave up refer to the same thing. Is that correct?

It'll be really great if someone could list out the fundamental differences between these various terms.

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Yes, convex and concave up mean the same thing.

The function $f(x)=\frac2x,x>0$ is strictly convex (or strictly concave up), because: $$f(tx_1+(1-t)x_2)< tf(x_1)+(1-t)f(x_2), \ \ \ 0<x_1<x_2,\ \ t\in [0,1] \ \ \text{or}\\ f'(x_1)< \frac{f(x_2)-f(x_1)}{x_2-x_1}, \ \ 0<x_1<x_2, \ \text{or}\\ f''(x)=\frac4{x^3}>0,x>0,$$ where $f\in C^0$, $f\in C^1$ or $f\in C^2$, respectively.

The function $f(x)=\frac2x$ is both quasi-concave and quasi-convex, because: $$f(tx_1+(1-t)x_2)\ge \min\{f(x_1),f(x_2)\}, t\in[0,1] \ \text{(quasi-concavity)}\\ f(tx_1+(1-t)x_2)\le \max\{f(x_1),f(x_2)\}, t\in[0,1] \ \text{(quasi-convexity)}$$

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