Property $D$ Functions

For a function $f(x)$ and a real number $a$, say $f(x)$ has property $D(a)$ if there exists $b \in \mathbb{R}$ such that

1. If $f(x) = \sin x$ has property $D(a)$, find all possible values of $a$. 2. Is there $a \in \mathbb{R}$ for which $f(x) = 2^x + 3^x$ has property $D(a)$? Explain. 3. Prove that no function $f(x)$ with domain $\mathbb{R}$ satisfies both of the following:

1. there is a unique $a \in \mathbb{R}$ for which $f(x)$ has property $D(a)$;
2. for every $q > 0$, the function $g(x) = f(x)q^x$ is monotonic on $\mathbb{R}$ (not necessarily strictly).