Finding f(0), f'(0), and f'(x) from functional equation and limit

[clarence]

Homework Statement

f(x+y)= f(x) + f(y) + y*x^2 +x*y^2 Given: lim of f(x)/x where x approaches 0 is 1
Find : 1) f(0) 2) f ' (0) 3) f ' (x)

Homework Equations

The Attempt at a Solution

countless hours with other coursemates that lead to nothing but a looping headache.

 

[statdad]

Note that

$$f(x+y) = f(x) + f(y) + x^2 y + x y^2$$

means that (using $$x = y = 0$$

$$f(0) = f(0) + f(0) + 0$$

What does this tell you about $$f(0)$$? Also remember the definition:

$$f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{(0+h) - 0} = \lim_{h \to 0} \frac{f(h)-f(0)} h$$

This second point relates to one thing you are given

 

[clarence]

Yep, that be true. It's the 3rd part of the question that i can't still solve. I've posted the first two parts of the question as a guide to solving the third part. So any ideas on the 3rd part? Thanks for making the second part ans more organize. My solution was a little more messy.

 

[clarence]

Note that

$$f(x+y) = f(x) + f(y) + x^2 y + x y^2$$

means that (using $$x = y = 0$$

$$f(0) = f(0) + f(0) + 0$$

What does this tell you about $$f(0)$$? Also remember the definition:

$$f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{(0+h) - 0} = \lim_{h \to 0} \frac{f(h)-f(0)} h$$

This second point relates to one thing you are given

but x+y=0, x can be -y. that is, x and y may not equal to zero?
so f(0)=f(x)+f(-x)?

 

[gabbagabbahey]

$$f'(x)= \lim_{h \to 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$

But given that $f(x+y)=f(x)+f(y)+x^2y+xy^2$ , what is $f(x+h)$ ?

 

[clarence]

Wow, true, I didn't see it that way.Thanks a lot. LOL