Suppose f′′ continuous in [a,b] check:

Question

I am solving some exercises of calculation book, and I'm stuck in this exercise, and I do not know how to make this exercise....

Suppose $f''$ continuous in $[a,b]$ check:

$$ f(b) = f(a) + f'(a)(b-a) + \int_{a}^{b}(b-t)f''(t)dt $$

Suppose $f'''$ continuous in $[a,b]$, Complete based on the previous question: $$ f(b) = f(a) + f'(a)(b-a) + \frac{f''(a)}{2}(b-a)^2 + \int_{a}^{b}\frac{(b-t)^2}{2}f'''(t)dt $$

I appreciate if someone help me with this exercise!

Answer 1

By parts:

$$\int_a^b(b-t)f''(t)dt=\left.(b-t)f'(t)\right|_a^b+\int_a^bf'(t)dt=(a-b)f'(a)+f(b)-f(a)$$

Now you try the other one.