Recent content by statsmichael

The book just says this function is concave, i.e. f(tx + (1-t)y) \geq tf(x) + (1-t)f(y) when t \in [0, 1], \forall x, y \in [0, \infty). That's concave downwards, right? Concave upwards is the same as convex, right? If the function can't be increasing, it can at least be non-decreasing, I think.

This is in an analysis textbook that I'm working through, and it mentions that since we haven't encountered differentiation in the course yet (it's in the section on basic metric spaces), I should be able to prove it without using anything besides proof by contradiction.

I thought the definition of a concave function was that any line connecting two points (x, f(x)) and (y, f(y)) will lie BELOW the graph. Isn't that the correct interpretation of the definition (see Wikipedia, for example: https://en.wikipedia.org/wiki/Concave_function) If I look at the line...

I'm a high school student working through some analysis and economics textbooks. I struggle a fair amount in math, which is why I sometimes post here and elsewhere looking for hints on problems or statements in my books that confuse me. Thank you!

Homework Statement My question is similar to this one (https://www.physicsforums.com/threads/metric-function-composed-with-concave-function.566338/), I think. I have a concave function f: [0, \infty) \rightarrow [0, \infty) where f(0) = 0, f(x) > 0, x \neq 0 and I need to show that this...