## suneja 3 years ago 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.

1. JakeV8

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.

2. suneja

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

3. JakeV8

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.

4. JakeV8

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).

5. suneja

ya upward parabola is concave up

6. suneja

concave fns hav minima ie second differenciation is >0

7. JakeV8

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

8. suneja

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

9. JakeV8

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.

10. suneja

ya.

11. JakeV8

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?

12. JakeV8

|dw:1348929716882:dw|

13. JakeV8

Do you think the made-up function in that diagram is considered concave but not differentiable because of the vertical slope piece in the middle? I don't really know...

14. JakeV8

I just looked it up... it doesn't have to be this hard :)

15. suneja

a fn is not differential at "kink" ie corners so it shudnt be diff

16. suneja

|dw:1348930186324:dw| it wont be diff der

17. JakeV8

an absolute value function is non-differentiable... f(x) = |x| is concave up... g(x) = |x + 2| is also... is f(x) + g(x) is concave?

18. JakeV8

|dw:1348930353864:dw|

19. suneja

k tis makes sense:)

20. JakeV8

Maybe... it's making my head hurt! with that absolute value example, does that h(x) qualify as concave up? It doesn't have a single minimum point, but the line across the bottom is a minimum area over that range, and the whole function does face up... just not sure about the definition of concave in a situation like this.

21. suneja

see a/c to the dat i hav they hav jst used tha fact dat f n g r concave all i want to knw is wat does tis phrase means " Neither fn is necessarily differentiable

22. JakeV8

I think it just means that you cannot rely on differentiating the functions as a way of showing that h(x) is concave. You have to be able to realize what adding concave functions does WITHOUT trying to find the derivative of h(x) to answer about its concavity.

23. suneja

maybe