What's wrong with just computing the integral?  Let $\theta = \tan^{-1} a$.

$$\begin{align*} F(a) &amp;= \Bigl[ a \sin x + \cos x \Bigr]_{x=0}^\theta + \Bigl[ -\cos x - a \sin x \Bigr]_{x=\theta}^{\pi/2} \\ &amp;= a \sin \theta + \cos \theta - 1 - a + \cos \theta + a \sin \theta \\ &amp;= 2(a \sin \theta + \cos \theta) - (a+1) \\ &amp;= 2 \sqrt{a^2+1} - (a+1), \quad \color{red}{a &gt; 0}. \end{align*}$$  Then differentiation yields $$F'(a) = \frac{2a}{\sqrt{a^2+1}} - 1,$$ for which there is a critical point at $a = 1/\sqrt{3}$.  We then need to verify that this indeed corresponds to a minimum.

Minimizing certain integral
There's an old card game I want, but it's been out of print for 15 years and currently going for >$1000 used. There are just 80 cards in the game, and I've got a friend who does own it.

My plan is just to scan the cards, print them two-sided, laminate them and cut them out with a knife. I could also get plastic pockets and print the front and back separately.

What are some common pitfalls in printing a card game, and what am I likely to have missed?

(Don't worry, of course I'll mail a fit donation to the manufacturers.)
