Inequalities

Can you build a distribution with the maximum theoretical spread?

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

An inequality involving integrals of squares of functions.

Generalise this inequality involving integrals.

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

Find the maximum value of n to the power 1/n and prove that it is a maximum.

Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.

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

Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.