Question

function f & g are both concave fns of a single variable. Neither fn is necessarily differentiable. is the fn defined by h(x)=f(x)+g(x) necessarily concave, necessarily convex or not necessarily either.

Answer 1

I am thinking it would be "necessarily concave"... what do you think?

I am still considering what impact comes from the fact that it says the functions f and g are not necessarily differentiable.

Answer 2

that wer my doubt area is.. cz wen u jst use the fact that f n g are concave u can show h is concave but wat difference does differentiability makes here

Answer 3

yes... interesting.

What would make f not differentiable but still allow it to be considered concave? There is the "easy" definition of concave as U shaped (bowl, open side up). But the definition based on derivatives depends, I would have thought, on the function being differentiable.

Maybe you can imagine a non-differentiable function that still faces upward.... if you can imagine one, then a second similar upward facing function probably (but I hate guessing!!) can just be added to the first one without affecting concavity.

Answer 4

certainly if you had f(x) = x^2 and g(x) = 2x^2, adding them to get h(x) = 3x^2 is still concave.

(wait, am I totally backwards on concave? an upward parabola is concave up, right? Not convex? It's been awhile since I've done this sort of problem).

Answer 5

ya upward parabola is concave up

Answer 6

concave fns hav minima ie second differenciation is >0

Answer 7

so, f(x) = x^2 is concave up, but it doesn't demonstrate non-differentiability.

But I brought it up as a simpler example... adding 2 concave-up parabolas results in a 3rd up-facing parabola

Answer 8

v need to prove tis ... how shud i go abt it

Answer 9

so, leaving aside the non-differentiable part for a sec, I'm pretty certain you could prove analytically that for any f(x) and g(x) that are concave up, h(x) is also concave up.

Answer 10

ya.

Answer 11

do you actually have to prove this? Or just answer? Also, help me on this non-differentiable idea... what makes something non-differentiable? Is it that a derivative is undefined, like a section of vertical slope?