Hi, I have been stuck on the following for ages, any guidance would be welcome.
A function f: R --> R is C^3 and,
f(a+h)= f(a) + f'(a + 0.5h)h
for real a and h 0
by applying Taylors theorem to f and f' show that the third
derivative of f is identically 0.
I can see that f'(a + 0.5h)h is the Lagrange error term
and I know that similarly f'(a+05h)=f'(a)+0.5f'' h
Therefore f(x)= f(a) + (f'(a) + 0.5 f''(a))(x-a).
Is this even the right way to attempt the question? If so, how does the above imply that the third derivative of f is always zero?