Rudin 8.10

[SOLVED] rudin 8.10

1. The problem statement, all variables and given/known data
Prove that [itex](1-x)^{-1} \leq \exp 2x[/tex] when [itex]0 \leq x \leq 1/2[/itex].

2. Relevant equations
[tex]e^x = \sum_{i=0}^{\infty}\frac{x^n}{n!}[/tex]

[tex]1/(1-x) = 1+x+x^2+\cdots[/tex]


3. The attempt at a solution
I tried working with the series and that failed miserable. Maybe I need to use calculus and find out whether the function is increasing or decreasing but I started that but I tried that a little and did not see how it would help.

 

They are both 1.

Let f(x) = 1/(1-x) and g(x) = e^{2x}.

Then [itex]f'(x) = 1/(1-x)^2[/itex] and [itex]g'(x) = 2 e^{2x}[/itex].

Is there an obvious inequality between f'(x) and g'(x) when [itex]0 \leq x \leq 1/2[/itex]? I do not see it.

 

Yay, I got it thanks!