Task 1a The log of $x^d \cdot f(x)$ function: $g(x) =\left(d - (log_2(x))^2\right) log_2(x) \equiv h(z)=\left(d - z^2\right) z$

The graph of $z$ vs $h(z)$ is a 3-degree polynomial with 3-degree coefficient -1, the right-most zero is $\sqrt{d}$. Accordingly, $\forall x>2^\sqrt{d} \rightarrow x^d \cdot f(x) < 1$ aka $f(x)$ is neg-liable

Task 1b Given $d$, then because $f(x)$ and $g(x)$ are neg-liable, $\exists a \forall x > a \rightarrow f(x) < x^{-(d+1)}$ and $\exists b \forall x > b \rightarrow g(x) < x^{-(d+1)}$

Therefore, $\forall x > max(a,b) \rightarrow f(x)+g(x) < 2 x^{-(d+1)}$

Because $\forall x>2 \rightarrow 2 x^{-(d+1)} < x^{-d}$, therefore $\forall x > max(2,a,b) \rightarrow f(x)+g(x) < 2 x^{-(d+1)} < x^{-d}$