1) Proove that for every x>=0 and 0<a<1

Question

x^a-ax<=1-a

2)prove that for 0<=x<=1:

x^m*(1-x)^n<= (m^m*n^n)/((m+n)^(m+n))

can someone please guide/help me in about how to solve and or approch these questions?

Answer 1

#1 I haven't followed the argument through yet, but I think if you play around with

0 < a < 1
(x+1)^a <= (x+1)^1
x^a + ax^(a-1) + ... + ax + 1 <= x+1

things will work out.