6. If $3 \sin ^{3} x+\cos ^{3} x=3$, then the value of $\sin ^{2018} x+\cos ^{2018} x$ is $\qquad$
Answer 1.
Analysis First, from $3 \sin ^{3} x+\cos ^{3} x=3$ we know that $\sin x \geq 0$ must hold, otherwise $\cos ^{3} x=3-3 \sin ^{3} x \geq 3$, which is a contradiction. Also, since $\sin ^{3} x \leq \sin ^{2} x, \cos ^{3} x \leq \cos ^{2} x$, we have
$$
3=3 \sin ^{3} x+\cos ^{3} x \leq 3 \sin ^{2} x+\cos ^{2} x=1+2 \sin ^{2} x
$$

Solving this, we get $\sin ^{2} x \geq 1$, so $\sin x=1$ and $\cos x=0$, thus $\sin ^{2018} x+\cos ^{2018} x=1$.