let $f(x)$ have three derivatives on $[0,a]$,and such $f(0)=f''(0)=0$,
show that: there exsit $\xi\in[0,a]$ such $$3af'(a)=3f(a)+a^3f'''(\xi)$$
I think we can use Tarlor to solve it, $$f(0)=f(a)+f'(a)(-a)+f''(a)/2\cdot a^2+\dfrac{f'''(\eta)}{6}(-a)^3$$ $$f''(0)=f''(a)+f'''(\eta_{2})(-a)$$ But this maybe not usefull,and I have try some old methods also can't prove this problem,can you help me，Thank you
