Inequalities

Generalise this inequality involving integrals.

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

Can you find a quadratic equation which passes close to these points?

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

Here the diagram says it all. Can you find the diagram?

An inequality involving integrals of squares of functions.