For a positive number \(a\), let \[I(a) = \int_0^a \! \left(4 - 2^{x^2} \right) \, dx.\]

Then \(\dfrac{dI}{da}=0\) when \(a\) equals

  1. \(\dfrac{1 + \sqrt{5}}{2}\),

  2. \(\sqrt{2}\),

  3. \(\dfrac{\sqrt{5} - 1}{2}\),

  4. \(1\).

We are looking for the stationary point of \(I\). What does this mean geometrically?

As \(a\) increases, for which points does \(I(a)\) increase and for which points does it decrease? What does this tell us about where the stationary points have to lie?