1. ## derivatives

Hi everyone.

I need to prove something and can't seem to make
heads or tails on how to start.

- f is a function in which f(0) = 0 and f(x)<=0 for every x.
if a,b>=0 prove that f(a+b)<=f(a)+f(b).

i hope i'm clear in my description.

2. 1. Since f " < 0 then f ' (x) is decreasing

2. define h(x) = f(x+b) -f(x)

3. h ' (x) = f '(x + b) - f ' (x) < 0 since f ' is decreasing

4. h(a) = f(a+b)- f(a)

h(0) = f(b)

h(a)- h(0) = f(a+b) - [ f(a) + f(b)]

5. by the MVT there is a number c in (0,a) st

a* h'(c) = h(a) -h(0) = f(a+b) - [ f(a) + f(b)] < 0

the result follows.

3. thanks!