Define $$F(a)=\int_0^{\pi/2}|\sin x-a\cos x|dx.$$
Find $a$ such that $F(a)$ is minimum.

My attempt is to use differentiation under integral sign. Namely, we have $$F(a)=\int_0^{\arctan a}(a\cos x-\sin x)dx+ \int_{\arctan a}^{\pi/2}(\sin x-a\cos x)dx,$$
and hence we can find the derivative as usual. Butthe calculation becomes hopelessly messy and I'm wondering is there any clearer solution. Thank you so much.


Question: What is a *slightly* less extreme equivalent to being "fluent" in a language?
Answer: If you are referring to a natural spoken language, one step down from fluent would be "conversant", such as "Fluent in French, conversant in Spanish."

For programming languages, I usually write something like: "Very experienced in C and C++, familiar with Python and C#". But I agree with Irwin that "proficient" seems to be one step below "fluent", so maybe it goes: fluent --> proficient --> familiar.