find a quadratic function f such that f(3)=33, f'(3)=22, f''(3)=8
start with $\displaystyle f(x) = ax^2 + bx + c$
since $\displaystyle f(3) = 33$ ... $\displaystyle 33 = a(9) + b(3) + c$
$\displaystyle f'(x) = 2ax + b$
since $\displaystyle f'(3) = 22$ ... $\displaystyle 22 = 2a(3) + b$
$\displaystyle f''(x) = 2a$
since $\displaystyle f''(3) = 8$ ... $\displaystyle 8 = 2a$
so ... find $\displaystyle a$, $\displaystyle b$, and $\displaystyle c$.