Method 1:$x^{\frac{2}{3}} + x^{\frac{1}{3}} -2$ ≤ 0
put x=8
$8^{\frac{2}{3}} + 8^{\frac{1}{3}} -2 =4$
and 4 ≰ 0
Hence,we can eliminate option (2),(4) and (5)
now put x=7
$7^{\frac{2}{3}} + 7^{\frac{1}{3}} -2$ ≰ 0
From here we can eliminate option (3) too.
Method 2: $x^{\frac{2}{3}} + x^{\frac{1}{3}} -2$ ≤ 0
we can rewrite this equation as
$(x^{\frac{1}{3}}-1) (x^{\frac{1}{3}}+2)$
-8 ≤ x ≤ 1
Hence,Option(1) -8 ≤ x ≤ 1 is the correct choice.