Shaw that $T(f)(x)=\int_0^x f(s)(1+s^2)ds$ is not surjective. Let $T:\mathcal C^1(\mathbb R)\longrightarrow \mathcal C^2(\mathbb R)$ the function defined by $$T(f)(x)=\int_0^x f(s)(1+s^2)ds$$
for all $f\in \mathcal C^1(\mathbb R)$. I have to show that $T$ is not surjective. How can I do ?

For every $f\in \mathcal{C}^1$,
$$T(f)(0)=\int_0^0 f(s)(1+s^2)ds=0 ;$$hence $T(f)$ cannot be a non-zero constant function.
