Option D should be Correct.
1) Put x = 3/4 in given equation i.e $x^{4} - 2ax^{2} + x + a^{2} - a$ , the new equation will become
$a^{2} - 2.125 a + 1.0664$
then find the roots of this quadratic equation it will be.
a= 0.812488 , a= 1.31251
you can see both values of a are greater than 3/4.
2) Now Put x = 2/3 in given equation i.e $x^{4} - 2ax^{2} + x + a^{2} - a$ the new equation will become
$a^{2} - 1.8888 a + 0.864197$
then find the roots of this quadratic equation.
a= 0.777984 , a= 1.11082
you can see both values of a are greater than 3/4
Hence D option is correct.