convexity of tangent function
We will show that the tangent function is convex on the interval .To do this, we will use the addition formula for the tangent
and the fact thata continuous
real function is convex (http://planetmath.org/ConvexFunction) if and only if .
We start with the observation that, if and , then by thearithmetic-geometric mean inequality (http://planetmath.org/ArithmeticGeometricMeansInequality),
so
Let and be two numbers in the interval . Set and .Then and By the addition formula, we have
Hence,
so the tangent function is convex.