You know that the value your function must be non-negative.
The total area under the curve must be 1.
Therefore, let a=0 and find b.
hmm, if i let a=0, i get the wrong answer straight away.
The book says: PI/2, 3*PI/4 for a and b respectively.
I tried solving the question and i got 0 <= x <= PI/4 -- which seems to be also correct, as the answer they give is basically exactly 1 period later.
Since 0 + PI/2 = PI/2, and PI/4 + PI/2 = 3*PI/4
So i think i basically got it right, just that my answer wasnt any of the multiple choices.
Btw, the way i solved it (which is basically what you said) was:
2sin(4x) >= 0
sin(4x) >= 0
4x >= asin(0)
4x >= 0, PI
x >= 0, PI/4
so 0 <= x <= PI/4
and i checked the integral from 0 - PI/4 of 2sin4x to make sure it == 1.