Determine the second derivative of $f(x)=-5x^3+17x^2-5x+3$.

This is a polynomial and we know how to find the derivative of a polynomial. Therefore,
$$f\,'(x)=-15x^2+34x-5.$$

Now $f\,'(x)$ is also a polynomial so we can find its derivative. It is
$$f\,''(x)=-30x+34.$$