How to find functions

F(x) = f(x) / g(x)

G(x) = f(x) * g(x)

so that the third derivative of F(x) and G(x) = 0.

d^3F(x)/dx^3=0

may i know functions for which the computational cost of evaluating their first derivative is higher than the function it self.

the taylor series of a function

g(x) = f(x + a * f(x)) around a point x = x0 is given as (hoping i am correct)

g(x) = f(x0) + f'(x0) * (x+a*f(x) -x0) + f''(x0)/2 * (x+a*f(x)-x0) + .... ------------(********)

but using the command

taylor(f(x+a*f(x)),x=x0,4)

gievs

f(x0+a*f(x0)) + f'(x0+a*f(x0)) * ....

 something which is different ...

how to get (***********)

suppose we want to find a curve

a * x * y + y + b * x + c = 0

given the following conditions:

y(x) = f(x)  and y'(x) = f'(x) and y(m)= f(m)

how to find a , b and c in terms of f, f'...

for which function f(x) the expression is minumum

(diff(f(x), x, x, x, x))*(diff(f(x), x, x))-(1/2)*(diff(f(x), x, x, x))^3