Proving f'''(x) = 0 using Taylor's Theorem

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In summary, the homework statement is that f is 3 times continuously differentiable, and by applying taylors theorem to f and f' show that the third derivative f''' of f is identically 0.
  • #1
stukbv
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Homework Statement



f is 3 times continuously differentiable,
f(a+h) = f(a) + f'(a+1/2*h)h whenever a is a real number and h>=0 .
By applying taylors theorem to f and f' show that the third derivative f''' of f is identically 0 ...
 
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  • #2
What have you tried already?
 
  • #3
I don't understand how to apply taylors theorem to this??!
 
  • #4
Apply it anyway, we'll see where we end up. Can you start by finding the Taylor expansion of f and f' around a? Then apply it to f(a+h) and f(a+h/2)...
 
  • #5
errmm f(x) = f(a) + f'(a)(x-a)?
 
  • #6
stukbv said:
errmm f(x) = f(a) + f'(a)(x-a)?

Take a term more! And don't gorget the remainder!

And what's the expansion of f' around a (two terms are enough now) and don't forget the remainder!
 
  • #7
ok so first one f(x)= f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2 + R3(x)
and f'(x)=f'(a) +f''(a)(x-a) + R2 ? - not sure never seen it applied to f' before?
 
  • #8
stukbv said:
ok so first one f(x)= f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2 + R3(x)
and f'(x)=f'(a) +f''(a)(x-a) + R2 ?

OK! You'll probably have to write out the remainders later on (that's where the f''' comes in)

Now, subsitute the f and f' in the equation f(a+h) = f(a) + f'(a+1/2*h)h by their Taylor expansions...

- not sure never seen it applied to f' before?

Well, it applies to all continuous differentiable functions, so certainly to f', no? :smile:
 
  • #9
ok done that and i get that R3 = r2 ?
 
  • #10
You should start by writing out the remainders...
 
  • #11
Ok now I've written out the remainders and i get that
f(a+h) = f(a) + f'(a)h + f''(a)h^2/2 + f'''(c)h^3/3!

f'(a+h/2) = f'(a) + f''(a)h/2 + f'''(a)h^2/4

so substituting them in i get
f(a) + f'(a)h + f''(a)h^2/2 + f'''(c)h^3/3! = f(a) +[ f'(a) + f''(a)h/2 + f'''(a)h^2/4]*h
^3=3f
when i cancel things i then get 2f'''(c)h^3=3f'''(c)h^3
Now what? i don't get it ?
 
  • #12
stukbv said:
Ok now I've written out the remainders and i get that
f(a+h) = f(a) + f'(a)h + f''(a)h^2/2 + f'''(c)h^3/3!

f'(a+h/2) = f'(a) + f''(a)h/2 + f'''(a)h^2/4

That should be f'''(d)h^2/4, with [itex]d\in [a,a+h/2][/itex]
[/QUOTE]

Thus

so substituting them in i get
f(a) + f'(a)h + f''(a)h^2/2 + f'''(c)h^3/3! = f(a) +[ f'(a) + f''(a)h/2 + f'''(a)h^2/4]*h
^3=3f
when i cancel things i then get 2f'''(c)h^3=3f'''(c)h^3

You should get (after cancelling the h): 2f'''(c)=3f'''(d), with [itex]c\in [a,a+h][/itex] and [itex]d\in [a,a+h/2][/itex].

Now, what happens if [itex]h\rightarrow 0[/itex]?
 
  • #13
Ah i see so we take different values to put into the remainders, i was unsure about that.
so do we get c and d are both a as h->0?
 
  • #14
stukbv said:
Ah i see so we take different values to put into the remainders, i was unsure about that.
so do we get c and d are both a as h->0?

Indeed, so in the limit, we get 2f'''(a)=3f'''(a), thus...
 
  • #15
it = 0! so obvious now, thanks a lot !
 

1. How does Taylor's Theorem relate to proving f'''(x) = 0?

Taylor's Theorem is a mathematical theorem that allows us to approximate a function using a series of polynomial terms. By using this theorem, we can show that if a function's third derivative, or f'''(x), is equal to 0, then the function is essentially a constant value. This is because the third derivative represents the rate of change of the second derivative, which in turn represents the rate of change of the original function. If the third derivative is 0, then the function is not changing at all, and therefore remains a constant.

2. What are the key components of using Taylor's Theorem to prove f'''(x) = 0?

The key components of using Taylor's Theorem to prove f'''(x) = 0 are the function itself, the point at which we are evaluating the function, and the number of terms in the polynomial approximation. Additionally, we must also have knowledge of the function's derivatives up to the third derivative at the point of evaluation.

3. Can Taylor's Theorem be used to prove f'''(x) = 0 for all functions?

No, Taylor's Theorem can only be used to prove that f'''(x) = 0 for analytic functions. An analytic function is one that can be expressed as a convergent power series, meaning that it can be approximated by a polynomial. Therefore, if a function is not analytic, Taylor's Theorem cannot be used to prove that its third derivative is equal to 0.

4. How can we use Taylor's Theorem to prove f'''(x) = 0 for a specific function?

To prove f'''(x) = 0 for a specific function, we must first determine the function's derivatives up to the third derivative at the point of evaluation. Then, we can use Taylor's Theorem to construct a polynomial approximation of the function, and show that the third derivative of this approximation is equal to 0. If this is the case, then we can conclude that f'''(x) = 0 for the original function.

5. What is the significance of proving f'''(x) = 0 using Taylor's Theorem?

Proving f'''(x) = 0 using Taylor's Theorem can be useful in many applications, as it allows us to approximate a function with a simpler, polynomial form. This can make it easier to analyze and understand the behavior of the function, and can also be used in numerical methods for solving differential equations. Additionally, it can also help us identify when a function is constant, which can have implications in fields such as physics and economics.

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