# Thread: first and second theorem

1. ## first and second theorem

Given a function g, continue to any x in R : g(1)=5 and the Integral of g(t)dt=2 in {0,1}

If f(x)=1/2(Integral of(x-t)^2)g(t)dt) in {0,x}

Prube that

f'(x)=x(integral(g(t)dt- t·g(t)dt) in {0,x}

then find f''(1) and f'''(1)

2. ## Re: first and second theorem

Hey nigromante.

Can you show us what you have tried (partial attempts are OK) as well as any ideas you have to solve the problem?

Hint: Use the fundamental theorem of calculus to link f(x) with f'(x).

3. ## Re: first and second theorem

Nailed it, I know this threads been dead, but I got stuck on it at first glance:

$\displaystyle f(x)=\frac{1}{2}\int _0^x(x^2-2xt+t^2)g(t)dt$ $\displaystyle =\frac{1}{2}[x^2\int _0^xg(t)dt-2x\int _0^xtg(t)dt+\int _0^xt^2g(t)dt]$

$\displaystyle f'(x)=\frac{1}{2}[2x\int _0^xg(t)dt+x^2g(x)-2\int _0^xtg(t)dt-2x^2g(x)+x^2g(x)]=x\int _0^xg(t)dt-\int _0^xtg(t)dt$

4. ## Re: first and second theorem

THanks, I get the same, but now I have a problem with the third derivate, on the second I get: f''(1)=Integrate of g(t)dt, {0,1}=2, but f'''(1) I think is zero, but it supose to be "5".