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.
