The smallest value of \[I(a) = \displaystyle\int_0^1 \! (x^2 - a)^2 \, \mathrm{d}x,\] as \(a\) varies, is

\(\frac{3}{20}\),

\(\frac{4}{45}\),

\(\frac{7}{13}\),

\(1\).

So the smallest possible value of \(I(a)\) is \(\frac{4}{45}\) (which occurs when \(a=\frac{1}{3}\)).

Hence the answer is (b).

Alternatively we can differentiate \[I(a) = \frac{1}{5} - \frac{2a}{3} + a^2\] with respect to \(a\), and put this equal to zero to get the same result.