6. Let $a b \neq 0, a^{2} \neq b^{2}$, find the solution to $a f(x)+b x^{2} f\left(\frac{1}{x}\right) \geqslant \sin x(x>0)$.
6. Let the function $p(x)\left(\forall p(x) \geqslant 0, x \in \mathbf{R}^{+}\right)$. Let
$$
a f(x)+b x^{2} f\left(\frac{1}{x}\right)=\sin x+p(x) \text {. }
$$

By substituting $x$ with $\frac{1}{x}$ in the above inequality, we get
$$
\text { af }\left(\frac{1}{x}\right)+b\left(\frac{1}{x}\right)^{2} f(x)=\sin \frac{1}{x}+p\left(\frac{1}{x}\right) \text {. }
$$

From (1) and (2), forming a system of equations and eliminating $f\left(\frac{1}{x}\right)$, we get
$$
f(x)=\frac{1}{a^{2}-b^{2}}\left[a \sin x-b x^{2} \sin \frac{1}{x}+a p(x)-b x^{2} p\left(\frac{1}{x}\right)\right], \forall p(x) \geqslant 0, x \in \mathbf{R}^{+} .
$$