Originally Posted by
elfenlied These are my bro's questions which are due on Thurs. Advanced thx for any help!
1. the goal of this exercise is to define 2^x from first principle.
Assume that 2^x is already defined when x=a/b is rational(e.g. as the ath power of the bth root of 2 when a,b>0), and that the usual rule 2^(x+y)=2^x * 2^y is valid for rational exponents. Let E(x)={2^r| r belongs to Q, r<x} (a) define 2^x=supE(x) and show that this is finite. (b) prove that 2^(x+y)=2^x * 2^y is valid for real exponents x,y.
2.Using the fundamental theorem of calculus show that (a) the only differentiable functions on R satisfying f'(x)=0, for all x, are constant functions. (b) the only twice differentiable functions on R satisfying f ' '(x)=0 are linear functions i.e function f(x)=ax+b for constant a, b.