Second Fundamental Theorem of Calculus... Let f have a continuous second derivative. Prove that f(x)=f(a)+(x−a)f′(a)+int ^x_a (x−t)f′′(t)dt. This is a modification of exercise 6.6.4 from Advanced Calculus by Fitzpatrick. I have seen that this question has been asked here: Proving f(x)=f(0)+f′(0)x+int ^x_0 (x−t)f′′(t)dt for all x. However, there didn't seem to be a suitable answer.

Andreasihf

Andreasihf

Answered question

2022-09-03

Second Fundamental Theorem of Calculus...
Let f have a continuous second derivative. Prove that
f ( x ) = f ( a ) + ( x a ) f ( a ) + a x ( x t ) f ( t ) d t .
Here is my attempt at the problem.
Since f has a continuous second derivative, then the first derivative is also continuous. Therefore, by the first fundamental theorem of calculus, we have that
f ( x ) = f ( a ) + a x f ( t ) d t .
Expanding out the right-hand side of the above using integration by parts, we see that
f ( x ) = f ( a ) + f ( t ) t a x t f ( t ) d t .
This is where I am confused.

Answer & Explanation

Skye Hamilton

Skye Hamilton

Beginner2022-09-04Added 14 answers

Let u ( t ) = f ( t ) and let v(t)=t−x (don't be confused by the fact that there's an x in the definition of v(t); right now we are keeping x fixed and varying t). Then, we apply integration by parts:
a x f ( t ) 1 d t = a x u ( t ) v ( t ) d t = u ( t ) v ( t ) | a x a x u ( t ) v ( t ) d t = ( f ( x ) 0 f ( a ) ( a x ) ) a x f ( t ) ( t x ) d t = ( x a ) f ( a ) + a x ( x t ) f ( t ) d t
Hence,
f ( x ) = f ( a ) + a x f ( t ) d t = f ( a ) + ( x a ) f ( a ) + a x ( x t ) f ( t ) d t

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