2) if the graph of f(x) is passing through (0,0) find the rule foor f(x)

1. Differentiate f(x) to get f'(x) = e^(kx) -x√2, i think is what you mean?

Set f'(x) = 0, x=1, and then solve for k,

Thus, e^(k) = √2

and log e^k = k = log e(√2) = √2

2. I'm not sure what you mean by rule floor, below i have integrated the function as that was in your description,

f(x) = [(1/k)*(e^(kx))]-[((x^2)√2)/2] + c where k is as above

using initial conditions, x=0, f(x)=y=0, we find c=-1/k

Hope this helps!