Let $\phi, a \in C^{\infty}([0,1])$ and assume $a(0)=1$. Suppose that $$ \int_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt =0 \qquad \text{for all $\tau \in \mathbb R$}. $$ Does it follow that $\phi$ is a constant in a neighborhood of $t=0$?

If the answer to the above question is affirmative I have a follow up question as follows: Suppose additionally that $b \in C^{\infty}([0,1]$ with $b(0)=1$ and instead of the latter equation there holds $$\Bigg |\int_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt\Bigg| \leq \frac{1}{|\tau|} \Bigg|\int_0^1 e^{\tau \,\phi(t)}\,b(t)\,dt\Bigg|$$ Does it again follow that $\phi(t)$ is constant in a neighborhood of $t=0$?