Do you mean that from f'(a+θh)=(f(a+h)-f(a))/h,
we get that f'(a)=(f(a+(1-θ)h)-f(a-θh))/h, f'(a+h)=(f(a+(2-θ)h)-f(a+(1-θ)h))/h
and than get f''(a) using l'hopital's rule? I doubt I'm doing what you mean, since it's leading me nowhere. Could you please give me some further hints?
Suppose that the conditions for the Mean Value Theorem hold for the function
f : [a, a + h] → R, so that for some θ ∈ (0, 1) we have f (a + h) − f (a) = hf ′ (a + θh).
Fix f and a, and for each non-zero h write θ(h) for a corresponding value of θ.
Prove that if f ′′ (a) exists and is non-zero...
Let L = {f } be a first-order language containing a unary function
symbol f , and no other non-logical symbols.
1.Write down a sentence χ of L which is satisfiable in some structure
with an infinite domain but is false in every structure with a finite domain.
What can you say about the size of...