## frx 2 years ago Show that $e ^{x} \ge 1+x$ for all real numbers x

1. frx

I have the idea that I could use the McLauren expansion for e^x $e ^{x}=\sum_{n=0}^{\infty}\frac{ x ^{n} }{ n! }$

2. frx

So the two first expansions would be $1+x$ but since I have to add the Lagrange reminder it would be proven bigger than 1+x

3. frx

Would that be somewhat correct?

4. frx

$e ^{x}=\sum_{n=0}^{1} \frac{ x ^{n} }{ n! }=1+x +\frac{ f ^{(n+1)}(\xi) }{(n+1)! }x ^{n+1}$ $e ^{x}=\sum_{n=0}^{1} \frac{ x ^{n} }{ n! }=1+x +\frac{ f ^{(2)}(\xi) }{(2)! }x ^{2}$ $f(x)=e ^{x}; f \prime(x)=e ^{x}; f \prime \prime(x)=e {^x}$ so that gives the remainder $\frac{ e ^{\xi}x ^{2}}{2! }$ and $e ^{x}=1+x+\frac{ e ^{\xi}x ^{2}}{2! }\ \ge 1+x$ Is this a genuine proof that$e^{x}\ge 1+x$ ?

5. frx

Oh forgot to write that $\xi \in (0,x)$

i think we can use by calculus principle...

differential calculus i meant

8. frx

What about the Mclaurinexpansion I made, doesn't it prove the fact that it's bigger?

sorry i forgot about Mclaurinexpansion, i f by differential i got it

10. frx

How did you show it using differentials?

e^x ≥ x + 1 e^x - 1 - x ≥ 0 let given f(x) = e^x - 1 - x, so f '(x) = e^x - 1 criticals points of f, hapended when saat f '(x) = 0 so, e^x - 1 = 0 get x = 0 to knowing the kind (max or min) of f, use the 2nd derivative so, f ''(x) = e^x for x=0, gives f ''(0) = 1 > 0 because f ''(0) > 0, therefore its kind is minimum the minimum value of f is f(0) = e^0 - 1 - 0 = 0 because f has the minimum value, is 0 so,, obviously f(x) ≥ 0 e^x - 1 - x ≥ 0 e^x ≥ x + 1 (proof) :)

12. frx

Oh that's a good way to show it, pretty simple to, thank you RadEn! :)