For all reals numbers x and h, let f be a continuous function such that

i) (f(x + h) = f(x) + f(h) + 2xh for all x and h

ii) lim as h approaches 0 of f(h) / h = 7

a) Find f(0). Justify

b) Use the definition of the derivative to find f'(x). Show your work.

http://imgur.com/a/ngOt2

Question

For all reals numbers x and h, let f be a continuous function such that

i) (f(x + h) = f(x) + f(h) + 2xh for all x and h

ii) lim as h approaches 0 of f(h) / h = 7

a) Find f(0). Justify

b) Use the definition of the derivative to find f'(x). Show your work.

http://imgur.com/a/ngOt2

phi · Answer

maybe let x=0 and h=0  so
f(0+0)= f(0) + f(0) + 2*0*0 = 2f(0)
f(0) = 2 f(0)
2 f(0) - f(0) = 0
(2-1) f(0) = 0
f(0) = 0
 ?

adamk · Answer

@phi Gotcha. I'm gonna give part b a try now.

phi · Answer

Loser posted a reply showing we only need to set h =0 and that also shows f(0)=0

adamk · Answer

Ah, ok. I thought I saw someone else. That makes sense. Is it wrong if I show x as 0 too?

adamk · Answer

@phi ^

phi · Answer

I don't think so, but the other way is nicer.