Let E = L-1 represent the inverse function of L, where L is defined in Exercise 5.3.7.
a) Use the Inverse Function Theorem to show that E is differentiable and strictly increasing on R with E'(x) = E (x), E(0) = 1, and E (1) = e.
b) Prove that E(x) †’ ˆž as x †’ ˆž and E(x) †’ 0 as x †’ -ˆž.
c) Prove that E(xq) = (E(x))q and E(q) = eq for all q ˆˆ Q and x ˆˆ R.
d) Prove that E(x + y) = E(x)E(y) for all x, y ˆˆ R.
e) For each a ˆˆ R define ea = E(a). Let x > 0 and define xa = ea log x: = E(aL(x)). prove that 0 0 and xa > ya for a

for all α, β ˆˆ R and x > 0.

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