Variational lemma - inequality version

Question

Suppose $f\in L^2(a,b)$, $a,b\in \mathbb{R}$ and $a<b$. How to prove (if it is possible) that

$$\int_{a}^{b}f(t)\varphi(t)dt\ge0\,\forall\varphi\in C^{\infty}_{0}(a,b)\,\text{and}\,\varphi\ge 0\Rightarrow f(t)\ge 0\,\,\text{a.e.}\, t\in(a,b)?$$